The Complexity of Large-scale Convex Programming under a Linear Optimization Oracle

Introduction

The last few years have seen an increasing interest in the application of convex programming (CP) models for machine learning, image processing, and polynomial optimization, etc. The CP problems arising from these applications, however, are often of high dimension and hence challenging to solve. In particular, they are generally beyond the capability of second-order interior-point methods due to the highly demanding iteration costs of these optimization techniques. This has motivated the currently active research on first-order methods which possess cheaper iteration costs for large-scale CP, including Nesterov’s optimal method Nest83-1; Nest04; Nest05-1 and several stochastic first-order algorithms in NJLS09-1; Lan10-3. These optimization algorithms are relatively simple, and suitable for the situation when low or moderate solution accuracy is sought-after.

In this paper, we study a different class of optimization algorithms, referred to as linear-optimization-based convex programming (LCP) methods, for large-scale CP. Specifically, consider the CP problem of

In particular, if $p$ is computed based on first-order information, then we call these algorithms first-order LCP methods. Clearly, the difference between first-order LCP methods and the more general first-order methods exists in the restrictions on the format of subproblems. For example, in the well-known subgradient (mirror) descent method nemyud:83 and Nesterov’s method Nest83-1; Nest04, we solve the projection (or prox-mapping) subproblems given in the form of

The development of LCP methods dates back to the conditional gradient (CndG) method (a.k.a., Frank-Wolfe algorithm) developed by FrankWolfe56-1 (see also Dunn79; Dunn80 for some earlier studies on this area). This method has recently regained some interests from both machine learning and optimization community (see, e.g., AhiTodd13-1; Bach12-1; BeckTeb04-1; CoxJudNem13-1; Clarkson10; Freund13; Hazan08; HarJudNem12-1; Jaggi11; Jaggi13; Jaggi10; LussTeb13-1; ShGosh11; ShenKim12-1) mainly for the following reasons.

Simplicity. The CndG method is simple to implement since it does not require the selection of the distance function $d(x)$ in (1.3) and the fine-tuning of stepsizes, which are required in most other first-order methods (with exceptions to some extent for a few level-type first-order methods, see BenNem05-1; Lan13-1). This property is also referred to affine invariance (see Jaggi13; AspJaggi13 for more discussions).

Structural properties for the generated solutions. The output solutions of the CndG method may have certain desirable structural properties, e.g., sparsity and low rank, as they can often be written as the convex combination of a small number of extreme points of $X$.

Numerical studies (e.g., HarJudNem12-1) indicate that the CndG method can be competitive to the more involved gradient-type methods for solving certain classes of CP problems. It is also worth noting that the CndG method is closely related to the von Neumann algorithm studied by Dantzig Dantzig91-1; Dantzig92-1, and later in Epelman and Freund EpelFreu00-1, which can be viewed as a specialized CndG method for solving linear/conic feasibility problems.

This paper focuses on the complexity analysis of CP under an ${\rm LO}$ oracle, as well as the development of new LCP methods for large-scale CP. In particular, we intend to provide a general framework for complexity studies for the LCP methods, by generalizing a few interesting complexity results existing in the literature (e.g., Clarkson10; HarJudNem12-1; Hazan08; Jaggi10; Jaggi13). Although there exists rich complexity theory for the general first-order methods for large-scale CP in the literature, the study on the complexity of CP under an ${\rm LO}$ oracle is still limited. More specifically, in view of the classic CP complexity theory nemyud:83; Nest04, if $f$ is a general nonsmooth Lipschitz continuous convex function such that

then the number of iterations required by any first-order methods to find an $\epsilon$-solution of (1.1), i.e., a point $\bar{x}\in X$ s.t. $f(\bar{x})-f^{*}\leq\epsilon$, cannot be smaller than ${\cal O}(1/\epsilon^{2})$ if $n$ is sufficiently large. In addition, if $f$ is a general smooth convex function satisfying

then the number of iterations required by any first-order methods to find an $\epsilon$-solution of (1.1) cannot be smaller than ${\cal O}(1/\sqrt{\epsilon})$ if $n$ is large enough. These lower complexity bounds can be achieved, for example, by the aforementioned subgradient (mirror) descend method and Nesterov’s method, respectively, for nonsmooth and smooth convex optimization. In addition, in a breakthrough paper, Nest05-1 studied an important class of saddle point problems with $f$ is given by

Our contribution in this paper lies on the following three aspects. Firstly, we establish a series of lower complexity bounds for solving different classes of CP problems under an ${\rm LO}$ oracle. In particular, we show that for solving general smooth CP problems satisfying (1.5), the complexity (or number of calls to the ${\rm LO}$ oracle), in the worst case, cannot be smaller than

where ${\cal O}(1)$ denotes an absolute constant, $n$ is the dimension of the problem, and $D_{X}:=\max_{x,y\in X}\|x-y\|$. It is worth noting that a similar lower bound has been established for the CndG method by Jaggi13. However, the lower bound in (1.7) shows explicitly the dependence on the dimension $n$, and the problem parameters $L$ and $D_{X}$. Moreover, for solving the aforementioned saddle point problems with $f$ given by (1.6), we show that the number of calls to the ${\rm LO}$ oracle cannot be smaller than

We further show that the number of calls to the ${\rm LO}$ oracle for solving general nonsmooth CP problems cannot be smaller than

It should be pointed out that these lower complexity bounds are obtained not only for the aforementioned first-order LCP methods, but also for any other LCP methods including those based on higher-order information.

Secondly, we formally establish the (near) optimality of the CndG method and its variants, in terms of the number of calls to the ${\rm LO}$ oracle, for solving different classes of CP problems under an ${\rm LO}$ oracle.

If $f$ is a smooth convex function satisfying (1.5), it is well-known that the number of iterations required by the classic CndG method to find an $\epsilon$-solution of (1.1) will be bounded by ${\cal O}(1/\epsilon)$ (see, e.g., Jaggi11; HarJudNem12-1; Jaggi13). Hence, in view of (1.7), the classic CndG is an optimal LCP method if $n$ is sufficiently large, i.e., $n\geq LD_{X}^{2}/\epsilon$. Moreover, it is also well-known that for general first-order methods, one can employ non-Euclidean norm $\|\cdot\|$ and the distance function $d(x)$ in (1.3) to accelerate the solutions for CP problems with certain types of feasible sets $X$. However, the CndG method is invariant to the selection of $\|\cdot\|$ and thus self-adaptive to the geometry of the feasible region $X$ (see also Jaggi11; Jaggi13).

If $f$ is a special nonsmooth function given by (1.6), we show that the CndG method can achieve the lower complexity bound in (1.8) after properly smoothing the objective function. Note that, although a similar bound has been developed in CoxJudNem13-1, the optimality of this bound has not yet been established. In addition, the smoothing technique developed here is slightly different from those in Nest05-1; CoxJudNem13-1 as we do not require explicit knowledge of $D_{X}$, $D_{Y}$ and the target accuracy $\epsilon$ given in advance.

If $f$ is a general nonsmooth function satisfying (1.4), we show that the CndG method can achieve a nearly optimal complexity bound in terms of its dependence on $\epsilon$ after properly incorporating the randomized smoothing technique (e.g., DuBaMaWa11). In particular, by applying this method to the bilinear saddle point problems with $f$ given by (1.6), we obtain an first-order algorithm which only requires linear optimization in both primal and dual space to solve this class of problems. It appears to us that no such techniques have been presented before in the literature (see discussions in Section 1 of Nest08-1).

We also discuss the possibility to improve the complexity of the CndG method under strong convexity assumption about $f(\cdot)$ and with an enhanced ${\rm LO}$ oracle (see also a related work by GarberHazan13).

Thirdly, we present a few new LCP methods, namely the primal averaging CndG (PA-CndG) and primal-dual averaging CndG (PDA-CndG) algorithms, for solving large-scale CP problems under an ${\rm LO}$ oracle. These methods are obtained by replacing the projection subproblems with linear optimization subproblems in Nesterov’s accelerated gradient methods. We demonstrate that these new LCP methods not only exhibit the aforementioned optimal (or nearly optimal) complexity bounds, in terms of the number of calls to the ${\rm LO}$ oracle, for solving different CP problems, but also possess some unique convergence properties. In particular, we show that the rate of convergence of these new LCP methods depends on the summation of the distances among the solutions of (1.2). By exploiting this fact, we develop certain necessary conditions for the ${\rm LO}$ oracle under which the PA-CndG and PDA-CndG would exhibit an ${\cal O}(1/\sqrt{\epsilon})$ iteration complexity for solving smooth CP problems. This result thus helps to build up the connection between LCP methods and the general optimal first-order methods for CP. We also demonstrate through our preliminary numerical experiments that one of these new methods, namely PDA-CndG, can significantly outperform the CndG method for solving certain classes of CP problems, e.g., those with box-type constraints.

This paper is organized as follows. We first introduce a few lower complexity bounds for solving different classes of CP problems under an ${\rm LO}$ oracle in Section 2. In Section 3, we formally show the optimality of the classic CndG method for solving smooth CP problems, develop different variants of the CndG method which are optimal or nearly optimal for solving different nonsmooth CP problems, and present possible improvement of the CndG method to solve strongly convex CP problems. We then present a few new LCP methods, namely PA-CndG and PDA-CndG, establish their convergence properties and conduct numerical comparisons in Section 4. Some brief concluding remarks are made in Section 5.

Notice that the constant $L$ in (1.5) and (1.13) depends on $\|\cdot\|$.

Lower Complexity Bounds for CP under an LO{\rm LO} oracle

Our goal in this section is to establish a few lower complexity bounds for solving different classes of CP problems under an ${\rm LO}$ oracle. More specifically, we first introduce a generic LCP algorithm in Subsection 2.1 and then present a few lower complexity bounds for these types of algorithms to solve different smooth and nonsmooth CP problems in Subsections 2.2 and 2.3, respectively.

The LCP algorithms solve problem (1.1) iteratively. In particular, at the $k$-th iteration, these algorithms perform a call to the ${\rm LO}$ oracle in order to update the iterates by minimizing a given linear function $\langle p_{k},x\rangle$ over the feasible region $X$. A generic framework for these types of algorithms is described as follows.

Observe the above LCP algorithm can be quite general. Firstly, there are no restrictions regarding the definition of the linear function $\langle p_{k},\cdot\rangle$. For example, if $f$ is a smooth function, then $p_{k}$ can be defined as the gradient computed at some feasible solution or a linear combination of some previously computed gradients. If $f$ is nonsmooth, we can define $p_{k}$ as the gradient computed for a certain approximation function of $f$. We can also consider the situation when some random noise or second-order information is incorporated into the definition of $p_{k}$. Secondly, the output solution $y_{k}$ is written as a convex combination of $x_{0},\ldots,x_{k}$, and thus can be different from any points in $\{x_{k}\}$. We will show in Sections 3 and 4 that Algorithm 1 covers, as certain special cases, the classic CndG method and several new LCP methods to be studied in this paper.

It is interesting to observe the difference between the above LCP algorithm and the general first-order methods for CP. One one hand, the LCP algorithm can only solve linear, rather than nonlinear subproblems (e.g., projection or prox-mapping) to update iterates. On the other hand, the LCP algorithm allows more flexibility in the definitions of the search direction $p_{k}$ and the output solution $y_{k}$.

2 Lower complexity bounds for smooth minimization

In this subsection, we consider a class of smooth CP problems, denoted by ${\cal F}^{1,1}_{L,\|\cdot\|}(X)$, which consist of any CP problems given in the form of (1.1) with $f$ satisfying assumption (1.5). Our goal is to derive a lower bound on the number of iterations required by any LCP methods for solving this class of problems.

The complexity analysis has been an important topic in convex programming (see nemyud:83; Nest04). However, the study on the complexity for LCP methods is quite limited. Existing results focus on a specific algorithm, namely the classic CndG method. More specifically, CanonCullum68-1 proved an asymptotic lower bound of $\Omega(1/k^{1+\mu})$, for any $\mu>0$, on the rate of convergence for the CndG method. Jaggi13 revisited this algorithm and established a lower bound on the number of iteration performed by this algorithm for finding an approximate solution with certain sparse pattern (see also Clarkson10 and Hazan08). Using the basic idea of Jaggi’s development, we provide lower complexity bounds which has explicit dependence on the problems dimension and other parameters, in addition to the target accuracy. Moreover, while the lower bounds in Jaggi13 were developed for smooth optimization problem, we also generalize these bound for nonsmooth and saddle point problems. for solving different classes of CP problem.

Similarly to the classic complexity analysis for CP in nemyud:83; Nest04, we assume that the ${\rm LO}$ oracle used in the LCP algorithm is resisting, implying that: i) the LCP algorithm does not know how the solution of (1.2) is computed; and ii) in the worst case, the ${\rm LO}$ oracle provides the least amount of information for the LCP algorithm to solve problem (1.1). Using this assumption, we will construct a class of worst-case instances in ${\cal F}^{1,1}_{L,\|\cdot\|}(X)$, inspired by Jaggi13, and establish a lower bound on the number of iterations required by any LCP algorithms to solve these instances.

Let $\epsilon>0$ be a given target accuracy. The number of iterations required by any LCP methods to solve the problem class ${\cal F}_{L,\|\cdot\|}^{1,1}(X)$, in the worst case, cannot be smaller than

Clearly, this class of problems belong to ${\cal F}_{L,\|\cdot\|}^{1,1}(X)$ with $\|\cdot\|=\|\cdot\|_{2}$.

Without loss of generality, we assume that the initial point is given by $x_{0}=De_{1}$ where $e_{1}=(1,0,\ldots,0)$ is the unit vector. Otherwise, for an arbitrary $x_{0}\in X_{0}$, we can consider a similar problem given by

and adapt our following argument to this problem without much modification.

Now suppose that problem (2.2) is to be solved by an LCP algorithm. At the $k$-th iteration, this algorithm will call the ${\rm LO}$ oracle to compute a new search point $x_{k}$ based on the input vector $p_{k}$, $k=1,\ldots$. We assume that the ${\rm LO}$ oracle is resisting in the sense that it always outputs an extreme point $x_{k}\in\{De_{1},De_{2},\ldots,De_{n}\}$ such that

Suppose that totally $q$ unit vectors from the set $\{e_{1},e_{p_{1}},e_{p_{2}},\ldots,e_{p_{k}}\}$ are linearly independent for some $1\leq q\leq k+1\leq n$. Without loss of generality, assume that the vectors $e_{1}$, $e_{p_{1}}$, $e_{p_{2}}$, …, $e_{p_{q-1}}$ are linearly independent. Therefore, we have

where the second identity follows from the definition of $f_{0}$ in (2.2). The above inequality together with (2.3) then imply that

By the definition of $D_{X}$ and $X_{0}$, and the fact that $\|\cdot\|=\|\cdot\|_{2}$, we can easily see that $D_{X_{0}}=\sqrt{2}D$ and hence that

Using (2.5) and the above identity, we conclude that, for any $1\leq k\leq\bar{K}$,

Our result then immediately follows since (2.2) is a special class of problems in ${\cal F}_{L,\|\cdot\|}^{1,1}(X)$.

We now add a few remarks about the results obtained in Theorem 2.1. First, it can be easily seen from (2.1) that, if $n\geq LD_{X}^{2}/(2\epsilon)$, then the number of iterations required by any LCP methods for solving ${\cal F}^{1,1}_{L,\|\cdot\|}(X)$, in the worst case, cannot be smaller than ${\cal O}(1)LD_{X}^{2}/\epsilon$. Second, it is worth noting that the objective function $f_{0}$ in (2.2) is actually strongly convex. Hence, the performance of the LCP methods, in terms of the number of calls to the ${\rm LO}$ oracle, cannot be improved by assuming strong convexity when $n$ is sufficiently large (see Section 3.4 for more discussions).

3 Lower complexity bounds for nonsmooth minimization

In this subsection, we consider two classes of nonsmooth CP problems. The first one is a general class of nonsmooth CP problems, denoted by ${\cal F}^{0}_{M,\|\cdot\|}(X)$, which consist of any CP problems given in the form of (1.1) with $f$ satisfying (1.4). The second one is a special class of bilinear saddle-point problems, denoted by ${\cal F}^{0}_{\|A\|}(X,Y)$, composed of all CP problems (1.1) with $f$ given by (1.6). Our goal in this subsection is to derive the lower complexity bounds for any LCP algorithms to solve these two classes of nonsmooth CP problems.

It can be seen that, if $f(\cdot)$ is given by (1.6), then

where $D_{Y}$ is given by (1.10). Hence, the saddle point problems ${\cal F}^{0}_{\|A\|}(X,Y)$ are a special class of nonsmooth CP problems.

Theorem 2.2 below provides a few lower complexity bounds for solving these two classes of nonsmooth CP problems by using LCP algorithms.

Let $\epsilon>0$ be a given target accuracy. Then, the number of iterations required by any LCP methods to solve the general nonsmooth problems ${\cal F}^{0}_{M,\|\cdot\|}(X)$ and saddle point problems ${\cal F}^{0}_{\|A\|}(X,Y)$, respectively, cannot be smaller than

where $D_{X}$ and $D_{Y}$ are defined in (1.10) and (1.11), respectively.

We first show the bound in (2.6). Consider the CP problem of

Clearly, this class of problems belong to ${\cal F}^{0}_{M,\|\cdot\|}(X)$ with $\|\cdot\|=\|\cdot\|_{2}$. Now suppose that problem (2.2) is to be solved by an arbitrary LCP method. Without loss of generality, we assume that the initial point is given by $x_{0}=De_{1}$ where $e_{1}=(1,0,\ldots,0)$ is the unit vector. Assume that the ${\rm LO}$ oracle is resisting in the sense that it always outputs an extreme point solution. By using an argument similar to the one used in the proof of (2.4), we can show that

where the identity follows from the definition of $\hat{f}_{0}$ in (2.8). The above inequality together with (2.9) then imply that

Using the above definition, (2.10) and the fact that $D_{X_{0}}=\sqrt{2}D$, we conclude that

for any $1\leq k\leq\bar{K}$. Our result in (2.6) then immediately follows since (2.8) is a special class of problems in ${\cal F}_{M,\|\cdot\|}^{0}(X)$.

In order to prove the lower complexity bound in (2.7), we consider a class of saddle point problems given in the form of

Clearly, these problems belong to ${\cal S}_{\|A\|}(X,Y)$ with $A=MI$. Noting that problem (2.11) is equivalent to

we can show the lower complexity bound in (2.7) by using an argument similar to the one used in the proof of bound (2.6).

Observe that while the lower complexity bound in (2.6) is in the same order of magnitude as the one established in nemyud:83 (see also Nest04) for the general first-order methods to solve ${\cal F}_{M,\|\cdot\|}^{0}(X)$. However, the bound in (2.6) holds not only for first-order LCP methods, but also for any other LCP methods, including those based on higher-order information to solve ${\cal F}_{M,\|\cdot\|}^{0}(X)$.

The Optimality of CndG Methods for CP under an LO{\rm LO} oracle

Our goal in this section is to establish the optimality or near optimality of the classic CndG method and its variants for solving different classes of CP problems under an ${\rm LO}$ oracle. More specifically, we discuss the classic CndG method for solving smooth CP problems ${\cal F}^{1,1}_{L,\|\cdot\|}(X)$ in Subsection 3.1, and then present different variants of the CndG method to solve general nonsmooth CP problems ${\cal F}^{0}_{\|A\|}(X,Y)$ and saddle point problems ${\cal F}^{0}_{M,\|\cdot\|}(X)$, respectively, in Subsections 3.2 and 3.3. Some discussions about strongly convex problems are included in Subsection 3.4.

The classic CndG method FrankWolfe56-1; DemRub70 is one of the earliest iterative algorithms to solve problem (1.1). The basic scheme of this algorithm is stated as follows.

We now add a few remarks about the classic CndG method. Firstly, it can be easily seen that the classic CndG method is a special case of the LCP algorithm discussed in Subsection 2.1. More specifically, the search direction $p_{k}$ appearing in the generic LCP algorithm is simply set to the gradient $f^{\prime}(y_{k-1})$ in Algorithm 2, and the output $y_{k}$ is taken as a convex combination of $y_{k-1}$ and $x_{k}$. Secondly, in order to guarantee the convergence of the classic CndG method, we need to properly specify the stepsizes $\alpha_{k}$ used in the definition of $y_{k}$. There are two popular options for selecting $\alpha_{k}$: one is to set

and the other is to compute $\alpha_{k}$ by solving a one-dimensional minimization problem:

It is well-known that if $f$ satisfies (1.5) and $\alpha_{k}$ is set to either (3.1) or (3.2), then the classic CndG method will exhibit an ${\cal O}(1/k)$ rate of convergence for solving problem (1.1) (see, e.g., Clarkson10; HarJudNem12-1; Hazan08; Jaggi10; Jaggi11; Jaggi13).

We now formally describe the convergence properties of the above classic CndG method. Observe that, in contrast with existing analysis of the classic CndG method, we state explicitly in Theorem 3.1 how the rate of convergence associated with this algorithm depends on distance between the previous iterate $y_{k-1}$ and the output of the ${\rm LO}$ oracle, i.e., $\|x_{k}-y_{k-1}\|$. In addition, our analysis for the classic CndG method is slightly different than the standard ones, and some of the techniques developed here will be used later for the analysis of some new LCP methods in Section 4. Also observe that, given a candidate solution $\bar{x}\in X$, we use the functional optimality gap $f(\bar{x})-f^{*}$ as a termination criterion for the algorithm. It is worth noting that Jaggi13 has recently showed that the CndG method also exhibit ${\cal O}(1/k)$ rate of convergence in terms of a stronger termination criterion, i.e., the Wolfe gap given by $\max_{x\in X}\langle f^{\prime}(\bar{x}),\bar{x}-x\rangle$, although Kha96 implicitly derived such bounds for the CndG method applied to the minimum volume covering ellipsoid problem. We also refer to HarJudNem12-1; Freund13 for some interesting convergence results for the CndG algorithm in terms of the latter termination criterion.

We first state a simple technical result.

Let $\gamma_{k}\in(0,1]$, $k=1,2,\ldots$, be given. If the sequence $\{\Delta_{k}\}_{k\geq 0}$ satisfies

Dividing both sides of (3.3) by $\Gamma_{k}$, we obtain

Summing up these inequalities, we obtain (3.4).

We are now ready to describe the main convergence properties of the CndG method.

Let $\{x_{k}\}$ be the sequence generated by the classic CndG method applied to problem (1.1) with the stepsize policy in (3.1) or (3.2). If $f(\cdot)$ satisfies (1.5), then for any $k=1,2,\ldots$,

Let $\Gamma_{k}$ be defined in (3.5) with

Subtracting $f(x)$ from both sides of the above inequality, we obtain

which, in view of Lemma 1, then implies that

where the last inequality follows from the fact that $\gamma_{1}=1$ and (3.8).

We now add a few remarks about the results obtained in Theorem 3.1. Firstly, note that by (3.6) and the definition of $D_{X}$ in (1.10), we have, for any $k=1,\ldots$,

Hence, the number of iterations required by the classic CndG method to find an $\epsilon$-solution of problem (1.1) is bounded by

Comparing the above bound with (2.1), we conclude that the classic CndG algorithm is an optimal LCP method for solving ${\cal F}_{L,\|\cdot\|}^{1,1}(X)$ if $n$ is sufficiently large.

Secondly, although the CndG method does not require the selection of the norm $\|\cdot\|$, the iteration complexity of this algorithm, as stated in (3.12), does depend on $\|\cdot\|$ as the two constants, i.e., $L\equiv L_{\|\cdot\|}$ and $D_{X}\equiv D_{X,\|\cdot\|}$, depend on $\|\cdot\|$ . However, since the result in (3.12) holds for an arbitrary $\|\cdot\|$, the iteration complexity of the classic CndG method to solve problem (1.1) can actually be bounded by

For example, if $X$ is a simplex, a widely-accepted strategy to accelerate gradient type methods is to set $\|\cdot\|=\|\cdot\|_{1}$ and $d(x)=\sum_{i=1}^{n}x_{i}\log x_{i}$ in (1.3), in order to obtain (nearly) dimension-independent complexity results, which only grow mildly with the increase of the dimension of the problem (see nemyud:83; Nest05-1; Lan10-3). On the other hand, the classic CndG method does automatically adjust to the geometry of the feasible set $X$ in order to obtain such scalability to high-dimensional problems (see Lemma 7 in Jaggi13 for some related discussions).

Thirdly, observe that the rate of convergence in (3.6) depends on $\|x_{k}-y_{k-1}\|$ which usually does not vanish as $k$ increases. For example, suppose $\{y_{k}\}\to x^{*}$ (this is true if $x^{*}$ is a unique optimal solution of (1.1)), the distance $\{\|x_{k}-y_{k-1}\|\}$ does not necessarily converge to zero unless $x^{*}$ is an extreme point of $X$. In these cases, the summation $\sum_{i=1}^{k}\|x_{i}-y_{i-1}\|^{2}$ increases linearly with respect $k$. We will discuss some techniques in Section 4 that might help to improve this situation.

2 Optimal CndG methods for saddle point problems under an LO{\rm LO} oracle

In this subsection, we show that the CndG method, after incorporating some proper modification, can achieve the optimal complexity for solving the saddle point problems ${\cal F}^{0}_{\|A\|}(X,Y)$ under an ${\rm LO}$ oracle.

let us denote $c_{v}:={\rm argmin}_{y\in Y}v(y)$, $V(y):=v(y)-v(c_{v})-\langle\nabla v(c_{v}),y-c_{v}\rangle$ and

Note that $V(y)$ is often referred to as the Bregman distance (from $y$ to $c_{v}$) in the literature, which was initially studied by Breg67 and later by many others (see AuTe06-1; BBC03-1; Kiw97-1; NJLS09-1 and references therein). Then the function $f(\cdot)$ in (1.6) can be closely approximated by

Indeed, by definition we have $0\leq V(y)\leq{\cal D}_{Y,V}^{2}$ and hence, for any $\eta\geq 0$,

Moreover, Nest05-1 shows that $f_{\eta}(\cdot)$ is differentiable and its gradients are Lipschitz continuous with the Lipschitz constant given by

In view of this result, we modify the CndG method to solve ${\cal F}^{0}_{\|A\|}(X,Y)$ by replacing the gradient $f^{\prime}(y_{k})$ in Algorithm 2 with the gradient $f^{\prime}_{\eta_{k}}(y_{k})$ for some $\eta_{k}>0$. Observe that in the original Nesterov’s smoothing scheme (Nest05-1), we first need to define the smooth approximation function $f_{\eta}$ in (3.16) by specifying in advance the smoothing parameter $\eta$ and then apply a smooth optimization method to solve the approximation problem. The specification of $\eta$ usually requires explicit knowledge of $D_{X}$, ${\cal D}_{Y,V}^{2}$ and the target accuracy $\epsilon$ given a priori. However, by using a novel analysis, we show that one can use variable smoothing parameters $\eta_{k}$ and thus does not need to know the target accuracy $\epsilon$ in advance. In addition, wrong estimation on $D_{X}$ and ${\cal D}_{Y,V}^{2}$ only affects the rate of convergence of the modified CndG method by a constant factor. Our analysis relies on a slightly different construction of $f_{\eta}(\cdot)$ in (3.16) (i.e., the constant term $\eta{\cal D}_{Y,V}^{2}$ in (3.16) does not appear in Nest05-1) and the following simple observation.

Let $f_{\eta}(\cdot)$ be defined in (3.16) and $\eta_{1}\geq\eta_{2}\geq 0$ be given. Then, we have $f_{\eta_{1}}(x)\geq f_{\eta_{2}}(x)$ for any $x\in X$.

The result directly follows from the definition of $f_{\eta}(\cdot)$ in (3.16) and the fact that $V(y)-{\cal D}_{Y,V}^{2}\leq 0$.

We are now ready to describe the main convergence properties of this modified CndG method to solve ${\cal F}^{0}_{\|A\|}(X,Y)$.

Let $\{x_{k}\}$ and $\{y_{k}\}$ be the two sequences generated by the CndG method with $f^{\prime}(y_{k})$ replaced by $f_{\eta_{k}}(y_{k})$, where $f_{\eta}(\cdot)$ is defined in (1.6). If the stepsizes $\alpha_{k}$, $k=1,2,\ldots$, are set to (3.1) or (3.2), and $\{\eta_{k}\}$ satisfies

where $D_{X}$ and ${\cal D}_{Y,V}$ are defined in (1.10) and (3.15), respectively.

Let $\Gamma_{k}$ and $\gamma_{k}$ be defined in (3.5) and (3.7), respectively. Similarly to (3.10), we have, for any $x\in X$,

where the second inequality follows from (3.19) and Lemma 2, and the third inequality follows from (3.17). Now subtracting $f(x)$ from both sides of the above inequality, we obtain, $\forall x\in X$,

which, in view of Lemma 1, (3.7) and (3.8), then implies that, $\forall x\in X$,

Our result in (3.20) then immediately follows from (3.17) and the above inequality. Now it is easy to see that the selection of $\eta_{k}$ in (3.21) satisfies (3.19). By (3.20) and (3.21), we have

where the last inequality follows from the fact that

A few remarks about the results obtained in Theorem 3.2 are in order. First, observe that the specification of $\eta_{k}$ in (3.21) requires the estimation of a few problem parameters, including $\|A\|$, $D_{X}$, ${\cal D}_{Y,V}$ and $\sigma_{v}$. However, wrong estimation on these parameters will only result in the increase on the rate of convergence of the modified CndG method by a constant factor. For example, if $\eta_{k}=1/\sqrt{k}$ for any $k\geq 1$, then (3.20) reduces to

It is worth noting that similar adaptive smoothing schemes can also be used when one applies Nesterov’s accelerated gradient method to solve ${\cal F}^{0}_{\|A\|}(X,Y)$. Second, suppose that the norm $\|\cdot\|$ in the dual space associated with $Y$ is an inner product norm and $v(y)=\|y\|^{2}/2$. In this case, by the definitions of $D_{Y}$ and ${\cal D}_{Y,V}$ in (1.11) and (3.15), we have ${\cal D}_{Y,V}\leq D_{Y}$. Using this observation and (3.22), we conclude that the number of iterations required by the modified CndG method to solve ${\cal F}^{0}_{\|A\|}(X,Y)$ can be bounded by

which, in view of Theorem 2.2, is optimal when $n$ is sufficiently large.

3 Nearly optimal CndG methods for general nonsmooth problems under an LO{\rm LO} oracle

In this subsection, we present a randomized CndG method and demonstrate that it can achieve a nearly optimal rate of convergence for solving general nonsmooth CP problems ${\cal F}^{0}_{M,\|\cdot\|}(X)$ under an ${\rm LO}$ oracle. To the best of our knowledge, no such CndG methods have not been presented before for solving general nonsmooth CP problems in the literature.

The basic idea is to approximate the general nonsmooth CP problems ${\cal F}^{0}_{M,\|\cdot\|}(X)$ by using the convolution-based smoothing. The intuition underlying such a approach is that convolving two functions yields a new function that is at least as smooth as the smoother one of the original two functions. In particular, let $\mu$ denote the density of a random variable with respect to Lebesgue measure and consider the function $f_{\mu}$ given by

where $Z$ is a random variable with density $\mu$. Since $\mu$ is a density with respect to Lebesgue measure, $f_{\mu}$ is differentiable (Bert73-1). The above convolution-based smoothing technique has been extensively studied in stochastic optimization, e.g., Bert73-1; DuBaMaWa11; KatKul72-1; Nest11-1; Rubin81. For the sake of simplicity, we assume throughout this subsection that $\|\cdot\|=\|\cdot\|_{2}$ and $Z$ is uniformly distributed over a certain Euclidean ball. The following result is known in the literature (see, e.g., DuBaMaWa11).

$f_{u}(x)$ has $M\sqrt{n}/u$-Lipschitz continuous gradient with respect to $\|\cdot\|_{2}$;

If $u_{1}\geq u_{2}\geq 0$, then $f_{u_{1}}(x)\geq f_{u_{2}}(x)$ for any $x\in X$.

Let $\{x_{k}\}$ and $\{y_{k}\}$ be the two sequences generated by the classic CndG method with $f^{\prime}(y_{k-1})$ replaced by the average of the sampled gradients, i.e.,

where $f_{u}$ is defined in (3.24) and $\{\xi_{1},\ldots,\xi_{T_{k}}\}$ is an i.i.d. sample of $\xi$. If the stepsizes $\alpha_{k}$, $k=1,2,\ldots$, are set to (3.1) or (3.2), and $\{u_{k}\}$ satisfies

where $M$ is given by (1.4). In particular, if

Let $\gamma_{k}$ be defined in (3.7), similarly to (3.9), we have

where the last inequality follows from Lemma 3.a), we conclude from (3.30) that, $\forall x\in X$,

Noting that by Jensen’s inequality and Lemma 3.c),

we conclude from the previous inequality that

which, in view of Lemma 1, (3.7) and (3.8), then implies that, $\forall x\in X$,

The result in (3.27) follows directly from Lemma 3.a) and the above inequality. Using (3.23), (3.27) and (3.28), we can easily verify that the bound in (3.29) holds.

We now add a few remarks about the results obtained in Theorem 3.3. Firstly, note that in order to obtain the result in (3.29), we need to set $T_{k}=k$. This implies that at the $k$-th iteration of the randomized CndG method in Theorem 3.3, we need to take an i.i.d. sample $\{\xi_{1},\ldots,\xi_{k}\}$ of $\xi$ and compute the corresponding gradients $\{f^{\prime}(y_{k-1},\xi_{1}),\ldots,f^{\prime}(y_{k-1},\xi_{k})\}$. Also note that from the proof of the above result, we can recycle the generated samples $\{\xi_{1},\ldots,\xi_{k}\}$ for usage in subsequent iterations.

Secondly, since ${\cal F}^{0}_{\|A\|}(X,Y)\subset{\cal F}^{0}_{M,\|\cdot\|}(X)$, we can apply the randomized CndG method to solve the saddle point problems ${\cal F}^{0}_{\|A\|}(X,Y)$. In comparison with the smoothing CndG method in Subsection 3.2, we do not need to solve the subproblems given in the form of (3.16), but to solve the subproblems

in order to compute $f^{\prime}(y_{k-1},\xi_{i})$, $i=1,\ldots,k$, at the $k$-th iteration. In particular, if $\hat{f}(y)=0$, then we only need to solve linear optimization subproblems over the set $Y$. To the best of our knowledge, this is the first time that optimization algorithms of this type has been proposed in the literature (see discussions in Section 1 of Nest08-1).

and that the total number of subgradient evaluations can be bounded by

According to the lower complexity bound in (2.6), we conclude that the above complexity bound in (3.32) is nearly optimal for the following reasons: i) the above result is in the the same order of magnitude as (2.6) with an additional factor of $\sqrt{n}$; and ii) the termination criterion is in terms of expectation. Note that while it is possible to show that the relation (3.29) holds with overwhelming probability by developing certain large deviation results associated with (3.29), such a result has been skipped in this paper for the sake of simplicity, see, e.g., GhaLan12 for some similar developments.

4 CndG methods for strongly convex problems under an enhanced LO{\rm LO} oracle

In this subsection, we assume that the objective function $f(\cdot)$ in (1.1) is smooth and strongly convex, i.e., in addition to (1.5), it also satisfies

These problems have been extensively studied in the literature. For example, it has been shown in Nest83-1; Nest04 that the optimal complexity for the general first-order methods to solve this class of problems is given by by

On the other hand, as noted in Subsection 2.2, the number of calls to the ${\rm LO}$ oracle for the LCP methods to solve these problems cannot be smaller than ${\cal O}(LD_{X}^{2}/\epsilon)$.

Our goal in this subsection is to show that, under certain stronger assumptions on the ${\rm LO}$ oracle, we can somehow “improve” the complexity of the CndG method for solving these strongly convex problems. More specifically, we assume throughout this subsection that we have access to an enhanced ${\rm LO}$ oracle, which can solve optimization problems given in the form of

for any given $x_{0}\in X$. For example, we can assume that the norm $\|\cdot\|$ is chosen such that problem (3.34) is relatively easy to solve. In particular, if $X$ is a polytope, we can set $\|\cdot\|=\|\cdot\|_{\infty}$ or $\|\cdot\|=\|\cdot\|_{1}$ and then the complexity to solve (3.34) will be comparable to the one to solve (1.2). Note however, that such a selection of $\|\cdot\|$ will possibly increase the value of the condition number given by $L/\mu$. Motivated by GhaLan10-1b, we present a shrinking CndG method under the above assumption on the enhanced ${\rm LO}$ oracle. It should be noted that the linear rate of convergence under stronger assumptions of the problem and/or the ${\rm LO}$ oracle is not completely new in the literature. More specifically, we notice that Garber and Hanzan GarberHazan13 have made some interesting development for the CndG methods applied to strongly convex problems, although the algorithm and analysis given here seem to be different from those in GarberHazan13. In addition, the linear convergence of the CndG method has been shown in PshDan78 for the case when the feasible set $X$ is round (strongly convex as a set), and similar result has been generalized in GuMar86 for the case when the optimal solution resides in the interior of the feasible set.

Note that an outer (resp., inner) iteration of the above shrinking CndG method occurs whenever $t$ (resp., $k$) increases by $1$. Observe also that the feasible set $X_{t}$ will be reduced at every outer iteration $t$. The following result summarizes the convergence properties for this algorithm.

Suppose that conditions (1.5) and (3.33) hold. If the stepsizes $\{\alpha_{k}\}$ in the shrinking CndG method are set to (3.1) or (3.2), then the number of calls to the enhanced ${\rm LO}$ oracle performed by this algorithm to find an $\epsilon$-solution of problem (1.1) can be bounded by

Denote $K\equiv 8L/\mu$. We first claim that $x^{*}\in X_{t}$ for any $t\geq 0$. This relation is obviously true for $t=0$ since $\|y_{0}-x^{*}\|\leq R_{0}=D_{X}$. Now suppose that $x^{*}\in X_{t-1}$ for some $t\geq 1$. Under this assumption, relation (3.11) holds with $x=x^{*}$ for inner iterations $k=1,\ldots,K$ performed at the $t$-th outer iteration. Hence, we have

Letting $k=K$ in the above relation, and using the facts that $p_{t}=y_{K}$ and $f(y_{K})-f^{*}\geq\mu\|y_{K}-x^{*}\|^{2}/2$, we conclude that

which implies that $x^{*}\in X_{t}$. We now provide a bound on the total number of calls to the ${\rm LO}$ oracle (i.e., the total number of inner iterations) performed by the shrinking CndG method. It follows from (3.37) and the definition of $R_{t}$ that

Hence the total number of outer iterations performed by the shrinking CndG method for finding an $\epsilon$-solution of (1.1) is bounded by $\lceil\max(\log\mu R_{0}/\epsilon,1)\rceil$. This observation, in view of the fact that $K$ inner iterations are performed at each outer iteration $t$, then implies that the total number of inner iterations is bounded by (3.35).

New variants of for LCP methods

Our goal in this section is to present a few new LCP methods for CP, obtained by replacing the projection (prox-mapping) subproblems with linear optimization subproblems in Nesterov’s accelerated gradient method. Throughout this section, we focus on smooth CP problems ${\cal F}^{1,1}_{L,\|\cdot\|}(X)$. However, the developed algorithms can be easily modified to solve saddle point problems, general nonsmooth CP problems and strongly convex problems, by using similar ideas to those described in Section 3.

In this subsection, we present a new LCP method, obtained by incorporating a primal averaging step into the CndG method. This algorithm is formally described as follows.

It can be easily seen that the PA-CndG method stated above is a special case of the LCP method in Algorithm 1. It differs from the classic CndG method in the way that the search direction $p_{k}$ is defined. In particular, while $p_{k}$ is set to $f^{\prime}(x_{k-1})$ in the classic CndG algorithm, the search direction $p_{k}$ in PA-CndG is given by $f^{\prime}(z_{k-1})$ for some $z_{k-1}\in{\rm Conv}\{x_{0},x_{1},\ldots,x_{k-1}\}$. In other words, we will need to “average” the primal sequence $\{x_{k}\}$ before calling the ${\rm LO}$ oracle to update the iterates. It is worth noting that the PA-CndG method can be viewed as a variant of Nesterov’s method in Nest04; Lan10-3, obtained by replacing the projection (or prox-mapping) subproblem with a simpler linear optimization subproblem.

By properly choosing the stepsize parameter $\alpha_{k}$, we have the following convergence results for the PA-CndG method described above.

Let $\{x_{k}\}$ and $\{y_{k}\}$ be the sequences generated by the PA-CndG method applied to problem (1.1) with the stepsize policy in (3.1) or (3.2). Then we have

Letting $l_{f}(\cdot,\cdot)$ be defined in (1.12), and using the previous two observations, (1.13), the definition of $x_{k}$ in Algorithm 4, and the convexity of $f(\cdot)$, we obtain

Subtracting $f(x)$ from both sides of the above inequality, we have

which, in view of Lemma 1, (3.8) and the fact that $\gamma_{1}=1$, then implies that, $\forall x\in X$,

We now add a few remarks about the results obtained in Theorem 4.1. Firstly, similarly to (3.12), we can easily see that the number of iterations required by the PA-CndG method to find an $\epsilon$-solution of problem (1.1) is bounded by ${\cal O}(1)LD_{X}^{2}/\epsilon$. Therefore, the PA-CndG method is an optimal LCP method for solving ${\cal F}^{1,1}_{L,\|\cdot\|}(X)$ when $n$ is sufficiently large. In addition, since the selection of $\|\cdot\|$ is arbitrary, the iteration complexity of this method can also be bounded by (3.13).

Secondly, while the rate of convergence for the CndG method (cf. (3.6)) depends on $\|x_{k}-y_{k-1}\|$, the one for the PA-CndG method depends on $\|x_{k}-x_{k-1}\|$, i.e., the distance between the output of the ${\rm LO}$ oracle in two consecutive iterations. Clearly, the distance $\|x_{k}-x_{k-1}\|$ will depend on the geometry of $X$ and the difference between $p_{k}$ and $p_{k-1}$. Let $\gamma_{k}$ be defined in (3.7) and suppose that $\alpha_{k}$ is set to (3.1) (i.e., $\alpha_{k}=\gamma_{k}$). Observe that by definitions of $z_{k}$ and $y_{k}$ in Algorithm 4, we have

which implies that $\|z_{k}-z_{k-1}\|\leq 3\gamma_{k}D_{X}$. Using this observation, (1.5) and the definition of $p_{k}$, we have

Hence, the difference between $p_{k}$ and $p_{k-1}$ vanishes as $k$ increases. By exploiting this fact, we establish in Corollary 1 certain necessary conditions about the ${\rm LO}$ oracle, under which the rate of convergence of the PA-CndG algorithm can be improved. It should be noted, however, that this result is more of theoretical interest only, since these assumptions on the ${\rm LO}$ oracle are quite strong and hard to be satisfied over a global scope.

Let $\{y_{k}\}$ be the sequence generated by the PA-CndG method applied to problem (1.1) with the stepsize policy in (3.1). Suppose that the ${\rm LO}$ oracle satisfies

for some $\rho\in(0,1]$ and $Q>0$. Then we have, for any $k\geq 1$,

Let $\gamma_{k}$ be defined in (3.7). By (4.3) and (4.4), we have

for any $k\geq 2$. The result follows by plugging the above bound into (4.1) and noting that

The bound obtained in (4.5) provides some interesting insights on the relation between first-order LCP methods and the general optimal first-order methods for CP. More specifically, if the ${\rm LO}$ oracle satisfies the Hölder’s continuity condition (4.4) for some $\rho\in(0.5,1]$, then we can obtain an ${\cal O}(1/k^{2})$ rate of convergence for the PA-CndG method for solving ${\cal F}^{1,1}_{L,\|\cdot\|}(X)$.

2 Primal-dual averaging CndG methods

Our goal in this subsection is to present another new LCP method, namely the primal-dual averaging CndG method, obtained by introducing a different acceleration scheme into the CndG method. This algorithm is formally described as follows.

Clearly, the above PDA-CndG method is also a special LCP algorithm. While the input vector $p_{k}$ to the ${\rm LO}$ oracle is set to $f^{\prime}(z_{k-1})$ in the PA-CndG method in the previous subsection, the vector $p_{k}$ in the PDA-CndG method is defined as a weighted average of $f^{\prime}(z_{i-1})$, $i=1,\ldots,k$, for some properly chosen weights $\theta_{i}$, $i=1,\ldots,k$. This algorithm can also be viewed as the projection-free version of an $\infty$-memory variant of Nesterov’s accelerated gradient method as stated in Nest05-1; tseng08-1.

Note that by convexity of $f$, the function $\Psi_{k}(x)$ given by

underestimates $f(x)$ for any $x\in X$. In particular, by the definition of $x_{k}$ in Algorithm 5, we have

and hence $\Psi_{k}(x_{k})$ provides a lower bound on the optimal value $f^{*}$ of problem (1.1). In order to establish the convergence of the PDA-CndG method, we first need to show a simple technical result about $\Psi_{k}(x_{k})$.

Let $\{x_{k}\}$ and $\{z_{k}\}$ be the two sequences computed by the PDA-CndG method. We have

where $l_{f}(\cdot\,;\cdot)$ and $\Psi_{k}(\cdot)$ are defined in (1.12) and (4.6), respectively.

It can be easily seen from (4.6) and the definition of $x_{k}$ in Algorithm 5 that $x_{k}\in{\rm Argmin}_{x\in X}\Psi_{k}(x)$ and hence that $\Psi_{k-1}(x_{k-1})\leq\Psi_{k-1}(x_{k})$. Using the previous observation and (4.6), we obtain

We are now ready to establish the main convergence properties of the PDA-CndG method.

Let $\{x_{k}\}$ and $\{y_{k}\}$ be the two sequences generated by the PDA-CndG method applied to problem (1.1) with the stepsize policy in (3.1) or (3.2). Also let $\{\gamma_{k}\}$ be defined in (3.7). If the parameters $\theta_{k}$ are chosen such that

for any $k=1,2,\ldots$, where $L$ is given by (1.13).

Using these two observations, (1.13), the definitions of $x_{k}$ in Algorithm 5, the convexity of $f$ and (4.8), we obtain

Also, using (4.9) and the fact that $\Theta_{k-1}=\Theta_{k}-\theta_{k}$, we have

Combining the above two relations and re-arranging the terms, we obtain

which, in view of Lemma 1, (3.7) and (3.8), then implies that

Our result then immediately follows from (4.7) and the above inequality.

We now add a few remarks about the results obtained in Theorem 4.2. Firstly, observe that we can simply set $\theta_{k}=k$, $k=1,2,\ldots$ in order to satisfy (4.9). Secondly, in view of the discussion after Theorem 4.1, the PDA-CndG method is also an optimal LCP method for ${\cal F}^{1,1}_{L,\|\cdot\|}(X)$ when $n$ is sufficiently large, since its the rate of convergence is exactly the same as the one for the PA-CndG method. In addition, its rate of convergence is invariant of the selection of the norm $\|\cdot\|$ (see (3.13)). Thirdly, according to (4.10), we can compute an online lower bound $\Psi_{k}(x_{k})$ on the optimal value $f^{*}$, and terminate the PDA-CndG method based on the optimality gap $f(y_{k})-\Psi_{k}(x_{k})$.

Similar to the PA-CndG method, the rate of convergence of the PDA-CndG method depends on $x_{k}-x_{k-1}$, which in turn depends on the geometry of $X$ and the input vectors $p_{k}$ and $p_{k-1}$ to the ${\rm LO}$ oracle. One can easily check the closeness between $p_{k}$ and $p_{k-1}$. Indeed, by the definition of $p_{k}$, we have $p_{k}=\Theta_{k}^{-1}[(1-\theta_{k})p_{k-1}+\theta_{k}f_{k}^{\prime}(z_{k-1})$ and hence

where the last inequality follows from (4.9). Noting that by (1.13), we have $\|f^{\prime}(x)\|_{*}\leq\|f^{\prime}(x^{*})\|_{*}+LD_{X}$ for any $x\in X$ and hence that $\|p_{k}\|_{*}\leq\|f^{\prime}(x^{*})\|_{*}+LD_{X}$ due to the definition of $p_{k}$. Using these observations, we obtain

Hence, under certain continuity assumptions on the ${\rm LO}$ oracle, we can obtain a result similar to Corollary 1. Note that both stepsize policies in (3.1) and (3.2) can be used in this result.

Let $\{y_{k}\}$ be the sequences generated by the PDA-CndG method applied to problem (1.1) with the stepsize policy in (3.1) or (3.2). Assume that (4.9) holds. Also suppose that the ${\rm LO}$ oracle satisfies (4.4) for some $\rho\in(0,1]$ and $Q>0$. Then we have, for any $k\geq 1$,

Similar to Corollary 1, Corollary 2 also helps to build some connections between LCP methods and the more general optimal first-order method. However, these results are more of theoretical interest only, since the ${\rm LO}$ oracle does not necessarily satisfy (4.4) for any $\rho>0$, but only for $\rho=0$ and $Q=D_{X}$.

3 Numerical Illustration

We can easily see that $\Delta_{n}\subset S_{n}$ by setting $x$ to be diagonal in (4.16). In our second set of experiments, we consider the QP problems over a hypercube, i.e., $\min_{x\in B_{n}}\|Ax-b\|_{2}^{2}$, where

It is well-known that to solve these problems becomes more and more difficult as $n$ increases. In our last set of experiments, we consider the QP problems over a hypercube intersected with a simplex, i.e., $\min_{x\in H_{n}(r)}\|Ax-b\|_{2}^{2}$, where

for some $r\in(0,1]$. These problems arise from certain important applications, including compressed sensing and portfolio optimization.

The CndG, PA-CndG and PDA-CndG algorithms are implemented in Matlab R2011b. Observe that we have discussed two stepsize policies for these algorithms: the stepsize policy (3.1) is the one that we have used in our experiments for its simplicity, while one can also use the stepsize policy (3.2) with more expensive iteration costs. The parameter $\theta_{k}$ in PDA-CndG is simply set to $\theta_{k}=k$. The initial point $y_{0}$ is randomly generated and remains the same for different algorithms. We report the results in Tables 2, 3 and 4, respectively, for minimization over simplex/spectrahedron, hypercube, and hypercube intersected with simplex. For each problem instance, we compute the objective values at the search points $y_{0}$, $y_{100}$ and $y_{1000}$, and the total CPU time (in seconds, Intel Core i7-2600 3.4 GHz) required for performing $1,000$ iterations of these algorithms.

We make a few observations about the results obtained in Tables 2, 3 and 4. Firstly, for solving the QP problems over a standard simplex/spectrahedron, all these three algorithms are about the same, with the CndG method slightly outperforming the other two. Secondly, for solving the QP problems over a hypercube, PDA-CndG can significantly outperform both CndG and PA-CndG by orders of magnitude. More specifically, as it can be seen from Table 3, although the CPU times for PDA-CndG are about as twice as the ones for CndG, the function values computed at the $100$ iterations of PDA-CndG are already comparable to those computed at the $1,000$ iterations for both CndG and PA-CndG. Moreover, the objective values at the $1,000$ iterations of the PDA-CndG are better than those for CndG and PA-CndG by $1-3$ accuracy digits, and the difference seems to become larger as $n$ increases. Thirdly, it can be seen from Table 4 that PDA-CndG also outperforms both CndG and PA-CndG by up to $2$ orders of magnitude for solving the QP problems over a hypercube intersected with simplex. Therefore, we conclude that the PDA-CndG method, although sharing similar worst-case complexity bounds with both CndG and PA-CndG, might significantly outperform the latter two algorithms for solving certain classes of CP problems, e.g., those with box-type constraints.

Concluding remarks

In this paper, we study a new class of optimization algorithms, namely the LCP methods, which covers the classic CndG method as a special case. We establish a few lower complexity bounds for these algorithms to solve different classes of CP problems. We formally show that the classic CndG method is an optimal LCP method for solving smooth CP problems and present new variants of this algorithm that are optimal or nearly optimal for solving certain saddle point and general nonsmooth problems under an ${\rm LO}$ oracle. Finally, we develop a few new LCP methods, namely PA-CndG and PDA-CndG, by properly modifying Nesterov’s accelerated gradient method, and show that they also exhibit the optimal rate of convergence for solving smooth CP problems under an ${\rm LO}$ oracle. In addition, we demonstrate through our preliminary numerical experiments that the PDA-CndG method can significantly outperform the classic CndG for solving certain classes of large-scale CP problems.

Acknowledgement

The author would like to thank Martin Jaggi and Elad Hazan for their help with improving the exposition of the paper and pointing out quite a few missing references in the original version of the paper. The author would also like to acknowledge support from the NSF grant CMMI-1000347, DMS-1319050, ONR grant N00014-13-1-0036 and NSF CAREER Award CMMI-1254446.

References