Structured Convex Optimization Research Articles

A Modified Primal-Dual Algorithm for Structured Convex Optimization with a Lipschitzian Term

A Modified Primal-Dual Algorithm for Structured Convex Optimization with a Lipschitzian Term

A hybrid alternating minimization algorithm for structured convex optimization problems with application in Poissonian image processing

<p style='text-indent:20px;'>Motivated by applications in image processing, we consider a class of structured convex optimization problems in which the objective function is the sum of two component functions with favorable structures, and, furthermore, both of which are composed with linear operators. In particular, to recover images degraded by Poisson noise, the famous Kullback-Leibler divergence can be adopted in data fitting, in which case one component function is separable, strongly convex and easy to minimize. The other component function usually allows an easily computable proximal operator. We then propose, analyze and test a hybrid algorithm particularly tailored for the underlying model structures, which can be viewed as a combination of the famous alternating minimization algorithm of Tseng and the alternating direction method of multipliers. Under mild conditions, global iterate convergence as well as ergodic <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{O}(1/N) $\end{document}</tex-math></inline-formula> sublinear convergence rate results measured by the function value residual and constraint violation of a reformulated constrained optimization problem are established, where <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula> denotes the iteration counter. The proposed algorithm is applied to fractional-order total variation regularized image denoising and deblurring problems, where the noise obeys Poisson distribution. Our experimental results demonstrate the effectiveness of the proposed algorithm.</p>

Accelerated gradient sliding for structured convex optimization

Our main goal in this paper is to show that one can skip gradient computations for gradient descent type methods applied to certain structured convex programming (CP) problems. To this end, we first present an accelerated gradient sliding (AGS) method for minimizing the summation of two smooth convex functions with different Lipschitz constants. We show that the AGS method can skip the gradient computation for one of these smooth components without slowing down the overall optimal rate of convergence. This result is much sharper than the classic black-box CP complexity results especially when the difference between the two Lipschitz constants associated with these components is large. We then consider an important class of bilinear saddle point problem whose objective function is given by the summation of a smooth component and a nonsmooth one with a bilinear saddle point structure. Using the aforementioned AGS method for smooth composite optimization and Nesterov’s smoothing technique, we show that one only needs \({{\mathcal{O}}}(1/\sqrt{\varepsilon })\) gradient computations for the smooth component while still preserving the optimal \({{\mathcal{O}}}(1/\varepsilon )\) overall iteration complexity for solving these saddle point problems. We demonstrate that even more significant savings on gradient computations can be obtained for strongly convex smooth and bilinear saddle point problems.

Primal-dual algorithms for multi-agent structured optimization over message-passing architectures with bounded communication delays

We consider algorithms for solving structured convex optimization problems over a network of agents with communication delays. It is assumed that each agent performs its local updates using possibly outdated information from its neighbours under the assumption that the delay with respect to each neighbour is bounded but otherwise arbitrary. The private objective of each agent is represented by the sum of two possibly nonsmooth functions one of which is composed with a linear mapping. The global optimization problem is the aggregate of local cost functions and a common Lipschitz-differentiable function. When the coupling between agents is represented only through the common function, the V\\~u-Condat primal-dual algorithm is studied. In the case when the linear maps introduce additional coupling between agents a new algorithm is developed. Moreover, a randomized variant of this algorithm is presented that allows the agents to wake up at random and independently from one another. The convergence of the proposed algorithms is established under strong convexity assumptions.

A Golden Ratio Primal–Dual Algorithm for Structured Convex Optimization

We design, analyze and test a golden ratio primal–dual algorithm (GRPDA) for solving structured convex optimization problem, where the objective function is the sum of two closed proper convex functions, one of which involves a composition with a linear transform. GRPDA preserves all the favorable features of the classical primal–dual algorithm (PDA), i.e., the primal and the dual variables are updated in a Gauss–Seidel manner, and the per iteration cost is dominated by the evaluation of the proximal point mappings of the two component functions and two matrix-vector multiplications. Compared with the classical PDA, which takes an extrapolation step, the novelty of GRPDA is that it is constructed based on a convex combination of essentially the whole iteration trajectory. We show that GRPDA converges within a broader range of parameters than the classical PDA, provided that the reciprocal of the convex combination parameter is bounded above by the golden ratio, which explains the name of the algorithm. An $$\mathcal {O}(1/N)$$ ergodic convergence rate result is also established based on the primal–dual gap function, where N denotes the number of iterations. When either the primal or the dual problem is strongly convex, an accelerated GRPDA is constructed to improve the ergodic convergence rate from $$\mathcal {O}(1/N)$$ to $$\mathcal {O}(1/N^2)$$ . Moreover, we show for regularized least-squares and linear equality constrained problems that the reciprocal of the convex combination parameter can be extended from the golden ratio to 2 and meanwhile a relaxation step can be taken. Our preliminary numerical results on LASSO, nonnegative least-squares and minimax matrix game problems, with comparisons to some state-of-the-art relative algorithms, demonstrate the efficiency of the proposed algorithms.

Variable Smoothing for Convex Optimization Problems Using Stochastic Gradients

We aim to solve a structured convex optimization problem, where a nonsmooth function is composed with a linear operator. When opting for full splitting schemes, usually, primal–dual type methods are employed as they are effective and also well studied. However, under the additional assumption of Lipschitz continuity of the nonsmooth function which is composed with the linear operator we can derive novel algorithms through regularization via the Moreau envelope. Furthermore, we tackle large scale problems by means of stochastic oracle calls, very similar to stochastic gradient techniques. Applications to total variational denoising and deblurring, and matrix factorization are provided.

GPU acceleration of ADMM for large-scale quadratic programming

The alternating direction method of multipliers (ADMM) is a powerful operator splitting technique for solving structured convex optimization problems. Due to its relatively low per-iteration computational cost and ability to exploit sparsity in the problem data, it is particularly suitable for large-scale optimization. However, the method may still take prohibitively long to compute solutions to very large problem instances. Although ADMM is known to be parallelizable, this feature is rarely exploited in real implementations. In this paper we exploit the parallel computing architecture of a graphics processing unit (GPU) to accelerate ADMM. We build our solver on top of OSQP, a state-of-the-art implementation of ADMM for quadratic programming. Our open-source CUDA C implementation has been tested on many large-scale problems and was shown to be up to two orders of magnitude faster than the CPU implementation.

A Generalized Alternating Linearization Bundle Method for Structured Convex Optimization with Inexact First-Order Oracles

In this paper, we consider a class of structured optimization problems whose objective function is the summation of two convex functions: f and h, which are not necessarily differentiable. We focus particularly on the case where the function f is general and its exact first-order information (function value and subgradient) may be difficult to obtain, while the function h is relatively simple. We propose a generalized alternating linearization bundle method for solving this class of problems, which can handle inexact first-order information of on-demand accuracy. The inexact information can be very general, which covers various oracles, such as inexact, partially inexact and asymptotically exact oracles, and so forth. At each iteration, the algorithm solves two interrelated subproblems: one aims to find the proximal point of the polyhedron model of f plus the linearization of h; the other aims to find the proximal point of the linearization of f plus h. We establish global convergence of the algorithm under different types of inexactness. Finally, some preliminary numerical results on a set of two-stage stochastic linear programming problems show that our method is very encouraging.

Generalized stochastic Frank–Wolfe algorithm with stochastic “substitute” gradient for structured convex optimization

The stochastic Frank–Wolfe method has recently attracted much general interest in the context of optimization for statistical and machine learning due to its ability to work with a more general feasible region. However, there has been a complexity gap in the dependence on the optimality tolerance $$\varepsilon $$ in the guaranteed convergence rate for stochastic Frank–Wolfe compared to its deterministic counterpart. In this work, we present a new generalized stochastic Frank–Wolfe method which closes this gap for the class of structured optimization problems encountered in statistical and machine learning characterized by empirical loss minimization with a certain type of “linear prediction” property (formally defined in the paper), which is typically present in loss minimization problems in practice. Our method also introduces the notion of a “substitute gradient” that is a not-necessarily-unbiased estimate of the gradient. We show that our new method is equivalent to a particular randomized coordinate mirror descent algorithm applied to the dual problem, which in turn provides a new interpretation of randomized dual coordinate descent in the primal space. Also, in the special case of a strongly convex regularizer our generalized stochastic Frank–Wolfe method (as well as the randomized dual coordinate descent method) exhibits linear convergence. Furthermore, we present computational experiments that indicate that our method outperforms other stochastic Frank–Wolfe methods for a sufficiently small optimality tolerance, consistent with the theory developed herein.

A linearized Peaceman–Rachford splitting method for structured convex optimization with application to stable principal component pursuit

Many applications arising from machine learning, statistics and image processing can be formulated as a convex minimization model with separable structures both in objective function and constraints. The Peaceman–Rachford splitting method is very efficient for solving these problems, but it is not convergent in the absence of some restrictive assumptions. In this paper, we propose a linearized Peaceman–Rachford splitting method by linearizing one subproblem. We analyze its convergence by proving the global convergence and establishing its worst-case convergence rate in the ergodic sense. Some randomly generated stable principal component pursuit problems are tested to illustrate the efficiency of the new algorithm.

SuperMann: A Superlinearly Convergent Algorithm for Finding Fixed Points of Nonexpansive Operators

Operator splitting techniques have recently gained popularity in convex optimization problems arising in various control fields. Being fixed-point iterations of nonexpansive operators, such methods suffer many well known downsides, which include high sensitivity to ill conditioning and parameter selection, and consequent low accuracy and robustness. As universal solution we propose SuperMann, a Newton-type algorithm for finding fixed points of nonexpansive operators. It generalizes the classical Krasnosel'skii-Mann scheme, enjoys its favorable global convergence properties and requires exactly the same oracle. It is based on a novel separating hyperplane projection tailored for nonexpansive mappings which makes it possible to include steps along any direction. In particular, when the directions satisfy a Dennis-Mor\'e condition we show that SuperMann converges superlinearly under mild assumptions, which, surprisingly, do not entail nonsingularity of the Jacobian at the solution but merely metric subregularity. As a result, SuperMann enhances and robustifies all operator splitting schemes for structured convex optimization, overcoming their well known sensitivity to ill conditioning.

The decompositions with respect to two core non-symmetric cones

It is known that the analysis to tackle with non-symmetric cone optimization is quite different from the way to deal with symmetric cone optimization due to the discrepancy between these types of cones. However, there are still common concepts for both optimization problems, for example, the decomposition with respect to the given cone, smooth and nonsmooth analysis for the associated conic function, conic-convexity, conic-monotonicity and etc. In this paper, motivated by Chares’s thesis (Cones and interior-point algorithms for structured convex optimization involving powers and exponentials, 2009), we consider the decomposition issue of two core non-symmetric cones, in which two types of decomposition formulae will be proposed, one is adapted from the well-known Moreau decomposition theorem and the other follows from geometry properties of the given cones. As a byproduct, we also establish the conic functions of these cones and generalize the power cone case to its high-dimensional counterpart.

Proximal Activation of Smooth Functions in Splitting Algorithms for Convex Image Recovery

Structured convex optimization problems typically involve a mix of smooth and nonsmooth functions. The common practice is to activate the smooth functions via their gradient and the nonsmooth ones ...

An Optimal Subgradient Algorithm with Subspace Search for Costly Convex Optimization Problems

This paper presents an acceleration of the optimal subgradient algorithm OSGA (Neumaier in Math Program 158(1–2):1–21, 2016) for solving structured convex optimization problems, where the objective function involves costly affine and cheap nonlinear terms. We combine OSGA with a multidimensional subspace search technique, which leads to a low-dimensional auxiliary problem that can be solved efficiently. Numerical results concerning some applications are reported. A software package implementing the new method is available.

Global Convergence of Unmodified 3-Block ADMM for a Class of Convex Minimization Problems

The alternating direction method of multipliers (ADMM) has been successfully applied to solve structured convex optimization problems due to its superior practical performance. The convergence properties of the 2-block ADMM have been studied extensively in the literature. Specifically, it has been proven that the 2-block ADMM globally converges for any penalty parameter $\gamma>0$. In this sense, the 2-block ADMM allows the parameter to be free, i.e., there is no need to restrict the value for the parameter when implementing this algorithm in order to ensure convergence. However, for the 3-block ADMM, Chen \etal \cite{Chen-admm-failure-2013} recently constructed a counter-example showing that it can diverge if no further condition is imposed. The existing results on studying further sufficient conditions on guaranteeing the convergence of the 3-block ADMM usually require $\gamma$ to be smaller than a certain bound, which is usually either difficult to compute or too small to make it a practical algorithm. In this paper, we show that the 3-block ADMM still globally converges with any penalty parameter $\gamma>0$ if the third function $f_3$ in the objective is smooth and strongly convex, and its condition number is in $[1,1.0798)$, besides some other mild conditions. This requirement covers an important class of problems to be called regularized least squares decomposition (RLSD) in this paper.

Non-asymptotic convergence analysis of inexact gradient methods for machine learning without strong convexity

Many recent applications in machine learning and data fitting call for the algorithmic solution of structured smooth convex optimization problems. Although the gradient descent method is a natural choice for this task, it requires exact gradient computations and hence can be inefficient when the problem size is large or the gradient is difficult to evaluate. Therefore, there has been much interest in inexact gradient methods (IGMs), in which an efficiently computable approximate gradient is used to perform the update in each iteration. Currently, non-asymptotic linear convergence results for IGMs are typically established under the assumption that the objective function is strongly convex, which is not satisfied in many applications of interest; while linear convergence results that do not require the strong convexity assumption are usually asymptotic in nature. In this paper, we combine the best of these two types of results by developing a framework for analysing the non-asymptotic convergence rates of IGMs when they are applied to a class of structured convex optimization problems that includes least squares regression and logistic regression. We then demonstrate the power of our framework by proving, in a unified manner, new linear convergence results for three recently proposed algorithms—the incremental gradient method with increasing sample size [R.H. Byrd, G.M. Chin, J. Nocedal, and Y. Wu, Sample size selection in optimization methods for machine learning, Math. Program. Ser. B 134 (2012), pp. 127–155; M.P. Friedlander and M. Schmidt, Hybrid deterministic–stochastic methods for data fitting, SIAM J. Sci. Comput. 34 (2012), pp. A1380–A1405], the stochastic variance-reduced gradient (SVRG) method [R. Johnson and T. Zhang, Accelerating stochastic gradient descent using predictive variance reduction, Advances in Neural Information Processing Systems 26: Proceedings of the 2013 Conference, 2013, pp. 315–323], and the incremental aggregated gradient (IAG) method [D. Blatt, A.O. Hero, and H. Gauchman, A convergent incremental gradient method with a constant step size, SIAM J. Optim. 18 (2007), pp. 29–51]. We believe that our techniques will find further applications in the non-asymptotic convergence analysis of other first-order methods.

SILVar: Single Index Latent Variable Models

A semi-parametric, non-linear regression model in the presence of latent variables is introduced. These latent variables can correspond to unmodeled phenomena or unmeasured agents in a complex networked system. This new formulation allows joint estimation of certain non-linearities in the system, the direct interactions between measured variables, and the effects of unmodeled elements on the observed system. The particular form of the model adopted is justified, and learning is posed as a regularized empirical risk minimization. This leads to classes of structured convex optimization problems with a "sparse plus low-rank" flavor. Relations between the proposed model and several common model paradigms, such as those of Robust Principal Component Analysis (PCA) and Vector Autoregression (VAR), are established. Particularly in the VAR setting, the low-rank contributions can come from broad trends exhibited in the time series. Details of the algorithm for learning the model are presented. Experiments demonstrate the performance of the model and the estimation algorithm on simulated and real data.

Asymmetric forward–backward–adjoint splitting for solving monotone inclusions involving three operators

In this work we propose a new splitting technique, namely Asymmetric Forward–Backward–Adjoint splitting, for solving monotone inclusions involving three terms, a maximally monotone, a cocoercive and a bounded linear operator. Our scheme can not be recovered from existing operator splitting methods, while classical methods like Douglas–Rachford and Forward–Backward splitting are special cases of the new algorithm. Asymmetric preconditioning is the main feature of Asymmetric Forward–Backward–Adjoint splitting, that allows us to unify, extend and shed light on the connections between many seemingly unrelated primal-dual algorithms for solving structured convex optimization problems proposed in recent years. One important special case leads to a Douglas–Rachford type scheme that includes a third cocoercive operator.

A unified approach to error bounds for structured convex optimization problems

Error bounds, which refer to inequalities that bound the distance of vectors in a test set to a given set by a residual function, have proven to be extremely useful in analyzing the convergence rates of a host of iterative methods for solving optimization problems. In this paper, we present a new framework for establishing error bounds for a class of structured convex optimization problems, in which the objective function is the sum of a smooth convex function and a general closed proper convex function. Such a class encapsulates not only fairly general constrained minimization problems but also various regularized loss minimization formulations in machine learning, signal processing, and statistics. Using our framework, we show that a number of existing error bound results can be recovered in a unified and transparent manner. To further demonstrate the power of our framework, we apply it to a class of nuclear-norm regularized loss minimization problems and establish a new error bound for this class under a strict complementarity-type regularity condition. We then complement this result by constructing an example to show that the said error bound could fail to hold without the regularity condition. Consequently, we obtain a rather complete answer to a question raised by Tseng. We believe that our approach will find further applications in the study of error bounds for structured convex optimization problems.

The restricted strong convexity revisited: analysis of equivalence to error bound and quadratic growth

The restricted strong convexity is an effective tool for deriving globally linear convergence rates of descent methods in convex minimization. Recently, the global error bound and quadratic growth properties appeared as new competitors. In this paper, with the help of Ekeland’s variational principle, we show the equivalence between these three notions. To deal with convex minimization over a closed convex set and structured convex optimization, we propose a group of modified versions and a group of extended versions of these three notions by using gradient mapping and proximal gradient mapping separately, and prove that the equivalence for the modified and extended versions still holds. Based on these equivalence notions, we establish new asymptotically linear convergence results for the proximal gradient method. Finally, we revisit the problem of minimizing the composition of an affine mapping with a strongly convex differentiable function over a polyhedral set, and obtain a strengthened property of the restricted strong convex type under mild assumptions.