Ergodic,primal convergence in dual subgradient schemes for convex programming,II: the case of inconsistent primal problems

Consider the utilization of a Lagrangian dual method which is convergent for consistent convex optimization problems. When it is used to solve an infeasible optimization problem, its inconsistency will then manifest itself through the divergence of the sequence of dual iterates. Will then the sequence of primal subproblem solutions still yield relevant information regarding the primal program? We answer this question in the affirmative for a convex program and an associated subgradient algorithm for its Lagrange dual. We show that the primal–dual pair of programs corresponding to an associated homogeneous dual function is in turn associated with a saddle-point problem, in which—in the inconsistent case—the primal part amounts to finding a solution in the primal space such that the Euclidean norm of the infeasibility in the relaxed constraints is minimized; the dual part amounts to identifying a feasible steepest ascent direction for the Lagrangian dual function. We present convergence results for a conditional \(\varepsilon \)-subgradient optimization algorithm applied to the Lagrangian dual problem, and the construction of an ergodic sequence of primal subproblem solutions; this composite algorithm yields convergence of the primal–dual sequence to the set of saddle-points of the associated homogeneous Lagrangian function; for linear programs, convergence to the subset in which the primal objective is at minimum is also achieved.     

Linear Convergence Rates for Variants of the Alternating Direction Method of Multipliers in Smooth Cases

In the present paper, we propose a novel convergence analysis of the alternating direction method of multipliers, based on its equivalence with the overrelaxed primal–dual hybrid gradient algorithm. We consider the smooth case, where the objective function can be decomposed into one differentiable with Lipschitz continuous gradient part and one strongly convex part. Under these hypotheses, a convergence proof with an optimal parameter choice is given for the primal–dual method, which leads to convergence results for the alternating direction method of multipliers. An accelerated variant of the latter, based on a parameter relaxation, is also proposed, which is shown to converge linearly with same asymptotic rate as the primal–dual algorithm.     

Relaxation methods for problems with strictly convex separable costs and linear constraints

We consider the minimization problem with strictly convex, possibly nondifferentiable, separable cost and linear constraints. The dual of this problem is an unconstrained minimization problem with differentiable cost which is well suited for solution by parallel methods based on Gauss-Seidel relaxation. We show that these methods yield the optimal primal solution and, under additional assumptions, an optimal dual solution. To do this it is necessary to extend the classical Gauss-Seidel convergence results because the dual cost may not be strictly convex, and may have unbounded level sets. Work supported by the National Science Foundation under grant NSF-ECS-3217668.     

Primal-dual row-action method for convex programming

We present a primal-dual row-action method for the minimization of a convex function subject to general convex constraints. Constraints are used one at a time, no changes are made in the constraint functions and their Jacobian matrix (thus, the row-action nature of the algorithm), and at each iteration a subproblem is solved consisting of minimization of the objective function subject to one or two linear equations. The algorithm generates two sequences: one of them, called primal, converges to the solution of the problem; the other one, called dual, approximates a vector of optimal KKT multipliers for the problem. We prove convergence of the primal sequence for general convex constraints. In the case of linear constraints, we prove that the primal sequence converges at least linearly and obtain as a consequence the convergence of the dual sequence.The research of the first author was partially supported by CNPq Grant No. 301280/86.     

Lower bounds and duality in the optimal control of distributed systems

We develop a duality theory for problems of minimization of a convex functional on a convex set and apply this theory to optimal control problems with non-differentiable functional and state as well as control constraints. The dual problem is sometimes found to be simpler than the primal problem. The equivalence of the primal and the dual problem is established for problems in which the functional is strongly convex. A posteriori error are also given for such problems.     

Application of the alternating direction method of multipliers to separable convex programming problems

This paper presents a decomposition algorithm for solving convex programming problems with separable structure. The algorithm is obtained through application of the alternating direction method of multipliers to the dual of the convex programming problem to be solved. In particular, the algorithm reduces to the ordinary method of multipliers when the problem is regarded as nonseparable. Under the assumption that both primal and dual problems have at least one solution and the solution set of the primal problem is bounded, global convergence of the algorithm is established.     

On certain optimization methods with finite-step inner algorithms for convex finite-dimensional problems with inequality constraints

Numerical methods are proposed for solving finite-dimensional convex problems with inequality constraints satisfying the Slater condition. A method based on solving the dual to the original regularized problem is proposed and justified for problems having a strictly uniformly convex sum of the objective function and the constraint functions. Conditions for the convergence of this method are derived, and convergence rate estimates are obtained for convergence with respect to the functional, convergence with respect to the argument to the set of optimizers, and convergence to the g-normal solution. For more general convex finite-dimensional minimization problems with inequality constraints, two methods with finite-step inner algorithms are proposed. The methods are based on the projected gradient and conditional gradient algorithms. The paper is focused on finite-dimensional problems obtained by approximating infinite-dimensional problems, in particular, optimal control problems for systems with lumped or distributed parameters.     

A proximal point algorithm revisit on the alternating direction method of multipliers

The alternating direction method of multipliers(ADMM)is a benchmark for solving convex programming problems with separable objective functions and linear constraints.In the literature it has been illustrated as an application of the proximal point algorithm(PPA)to the dual problem of the model under consideration.This paper shows that ADMM can also be regarded as an application of PPA to the primal model with a customized choice of the proximal parameter.This primal illustration of ADMM is thus complemental to its dual illustration in the literature.This PPA revisit on ADMM from the primal perspective also enables us to recover the generalized ADMM proposed by Eckstein and Bertsekas easily.A worst-case O(1/t)convergence rate in ergodic sense is established for a slight extension of Eckstein and Bertsekas’s generalized ADMM.     

Dual Convergence for Penalty Algorithms in Convex Programming

Algorithms for convex programming, based on penalty methods, can be designed to have good primal convergence properties even without uniqueness of optimal solutions. Taking primal convergence for granted, in this paper we investigate the asymptotic behavior of an appropriate dual sequence obtained directly from primal iterates. First, under mild hypotheses, which include the standard Slater condition but neither strict complementarity nor second-order conditions, we show that this dual sequence is bounded and also, each cluster point belongs to the set of Karush–Kuhn–Tucker multipliers. Then we identify a general condition on the behavior of the generated primal objective values that ensures the full convergence of the dual sequence to a specific multiplier. This dual limit depends only on the particular penalty scheme used by the algorithm. Finally, we apply this approach to prove the first general dual convergence result of this kind for penalty-proximal algorithms in a nonlinear setting.     

Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers

In this paper, under the existence of a certificate of nonnegativity of the objective function over the given constraint set, we present saddle-point global optimality conditions and a generalized Lagrangian duality theorem for (not necessarily convex) polynomial optimization problems, where the Lagrange multipliers are polynomials. We show that the nonnegativity certificate together with the archimedean condition guarantees that the values of the Lasserre hierarchy of semidefinite programming (SDP) relaxations of the primal polynomial problem converge asymptotically to the common primal–dual value. We then show that the known regularity conditions that guarantee finite convergence of the Lasserre hierarchy also ensure that the nonnegativity certificate holds and the values of the SDP relaxations converge finitely to the common primal–dual value. Finally, we provide classes of nonconvex polynomial optimization problems for which the Slater condition guarantees the required nonnegativity certificate and the common primal–dual value with constant multipliers and the dual problems can be reformulated as semidefinite programs. These classes include some separable polynomial programs and quadratic optimization problems with quadratic constraints that admit certain hidden convexity. We also give several numerical examples that illustrate our results.     

Penalty and Barrier Methods for Convex Semidefinite Programming

In this paper we present penalty and barrier methods for solving general convex semidefinite programming problems. More precisely, the constraint set is described by a convex operator that takes its values in the cone of negative semidefinite symmetric matrices. This class of methods is an extension of penalty and barrier methods for convex optimization to this setting. We provide implementable stopping rules and prove the convergence of the primal and dual paths obtained by these methods under minimal assumptions. The two parameters approach for penalty methods is also extended. As for usual convex programming, we prove that after a finite number of steps all iterates will be feasible.     

First-order methods of smooth convex optimization with inexact oracle

We introduce the notion of inexact first-order oracle and analyze the behavior of several first-order methods of smooth convex optimization used with such an oracle. This notion of inexact oracle naturally appears in the context of smoothing techniques, Moreau–Yosida regularization, Augmented Lagrangians and many other situations. We derive complexity estimates for primal, dual and fast gradient methods, and study in particular their dependence on the accuracy of the oracle and the desired accuracy of the objective function. We observe that the superiority of fast gradient methods over the classical ones is no longer absolute when an inexact oracle is used. We prove that, contrary to simple gradient schemes, fast gradient methods must necessarily suffer from error accumulation. Finally, we show that the notion of inexact oracle allows the application of first-order methods of smooth convex optimization to solve non-smooth or weakly smooth convex problems.     

A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms

We propose a new first-order splitting algorithm for solving jointly the primal and dual formulations of large-scale convex minimization problems involving the sum of a smooth function with Lipschitzian gradient, a nonsmooth proximable function, and linear composite functions. This is a full splitting approach, in the sense that the gradient and the linear operators involved are applied explicitly without any inversion, while the nonsmooth functions are processed individually via their proximity operators. This work brings together and notably extends several classical splitting schemes, like the forward–backward and Douglas–Rachford methods, as well as the recent primal–dual method of Chambolle and Pock designed for problems with linear composite terms.     

求解两块可分离凸极小化问题的惯性交替方向乘子法

The alternating direction method of multipliers(ADMM)is a widely used method for solving many convex minimization models arising in signal and image processing.In this paper,we propose an inertial ADMM for solving a two-block separable convex minimization problem with linear equality constraints.This algorithm is obtained by making use of the inertial Douglas-Rachford splitting algorithm to the corresponding dual of the primal problem.We study the convergence analysis of the proposed algorithm in infinite-dimensional Hilbert spaces.Furthermore,we apply the proposed algorithm on the robust principal component analysis problem and also compare it with other state-of-the-art algorithms.Numerical results demonstrate the advantage of the proposed algorithm.     

Dual Semidefinite Programs Without Duality Gaps for a Class of Convex Minimax Programs

In this paper, we introduce a new dual program, which is representable as a semidefinite linear programming problem, for a primal convex minimax programming problem, and we show that there is no duality gap between the primal and the dual whenever the functions involved are sum-of-squares convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust sum-of-squares convex programming problems under data uncertainty and to minimax fractional programming problems with sum-of-squares convex polynomials. We obtain these results by first establishing sum-of-squares polynomial representations of non-negativity of a convex max function over a system of sum-of-squares convex constraints. The new class of sum-of-squares convex polynomials is an important subclass of convex polynomials and it includes convex quadratic functions and separable convex polynomials. The sum-of-squares convexity of polynomials can numerically be checked by solving semidefinite programming problems whereas numerically verifying convexity of polynomials is generally very hard.     

Analysis of some interior point continuous trajectories for convex programming

In this paper, we analyse three interior point continuous trajectories for convex programming with general linear constraints. The three continuous trajectories are derived from the primal–dual path-following method, the primal–dual affine scaling method and the central path, respectively. Theoretical properties of the three interior point continuous trajectories are fully studied. The optimality and convergence of all three interior point continuous trajectories are obtained for any interior feasible point under some mild conditions. In particular, with proper choice of some parameters, the convergence for all three interior point continuous trajectories does not require the strict complementarity or the analyticity of the objective function. These results are new in the literature.     

On the convergence of the entropy-exponential penalty trajectories and generalized proximal point methods in semidefinite optimization

The convergence of primal and dual central paths associated to entropy and exponential functions, respectively, for semidefinite programming problem are studied in this paper. It is proved that the primal path converges to the analytic center of the primal optimal set with respect to the entropy function, the dual path converges to a point in the dual optimal set and the primal-dual path associated to this paths converges to a point in the primal-dual optimal set. As an application, the generalized proximal point method with the Kullback-Leibler distance applied to semidefinite programming problems is considered. The convergence of the primal proximal sequence to the analytic center of the primal optimal set with respect to the entropy function is established and the convergence of a particular weighted dual proximal sequence to a point in the dual optimal set is obtained.     

Least-norm linear programming solution as an unconstrained minimization problem

It is shown that the dual of the problem of minimizing the 2-norm of the primal and dual optimal variables and slacks of a linear program can be transformed into an unconstrained minimization of a convex parameter-free globally differentiable piecewise quadratic function with a Lipschitz continuous gradient. If the slacks are not included in the norm minimization, one obtains a minimization problem with a convex parameter-free quadratic objective function subject to nonnegativity constraints only.     

Subgradient Methods for Saddle-Point Problems

We study subgradient methods for computing the saddle points of a convex-concave function. Our motivation comes from networking applications where dual and primal-dual subgradient methods have attracted much attention in the design of decentralized network protocols. We first present a subgradient algorithm for generating approximate saddle points and provide per-iteration convergence rate estimates on the constructed solutions. We then focus on Lagrangian duality, where we consider a convex primal optimization problem and its Lagrangian dual problem, and generate approximate primal-dual optimal solutions as approximate saddle points of the Lagrangian function. We present a variation of our subgradient method under the Slater constraint qualification and provide stronger estimates on the convergence rate of the generated primal sequences. In particular, we provide bounds on the amount of feasibility violation and on the primal objective function values at the approximate solutions. Our algorithm is particularly well-suited for problems where the subgradient of the dual function cannot be evaluated easily (equivalently, the minimum of the Lagrangian function at a dual solution cannot be computed efficiently), thus impeding the use of dual subgradient methods.     

Augmented Lagrangian algorithms for linear programming

We introduce new augmented Lagrangian algorithms for linear programming which provide faster global convergence rates than the augmented algorithm of Polyak and Treti'akov. Our algorithm shares the same properties as the Polyak-Treti'akov algorithm in that it terminates in finitely many iterations and obtains both primal and dual optimal solutions. We present an implementable version of the algorithm which requires only approximate minimization at each iteration. We provide a global convergence rate for this version of the algorithm and show that the primal and dual points generated by the algorithm converge to the primal and dual optimal set, respectively.