Nonlinear Proximal Decomposition Method for Convex Programming

In this paper, we propose a new decomposition method for solving convex programming problems with separable structure. The proposed method is based on the decomposition method proposed by Chen and Teboulle and the nonlinear proximal point algorithm using the Bregman function. An advantage of the proposed method is that, by a suitable choice of the Bregman function, each subproblem becomes essentially the unconstrained minimization of a finite-valued convex function. Under appropriate assumptions, the method is globally convergent to a solution of the problem.     

Some properties of generalized proximal point methods for quadratic and linear programming

The proximal point method for convex optimization has been extended recently through the use of generalized distances (e.g., Bregman distances) instead of the Euclidean one. One advantage of these extensions is the possibility of eliminating certain constraints (mainly positivity) from the subproblems, transforming an inequality constrained problem into a sequence of unconstrained or equality constrained problems. We consider here methods obtained using a certain class of Bregman functions applied to convex quadratic (including linear) programming, which are of the interior-point type. We prove that the limit of the sequence generated by the method lies in the relative interior of the solution set, and furthermore is the closest optimal solution to the initial point, in the sense of the Bregman distance. These results do not hold for the standard proximal point method, i.e., when the square of the Euclidean norm is used as the Bregman distance.The research leading to this paper was partially supported by CNPq Grant 301280/86.We thank two anonymous referees whose comments and suggestions allowed us to improve our results in a significant way.     

An interior proximal linearized method for DC programming based on Bregman distance or second-order homogeneous kernels

Abstract

We present an interior proximal method for solving constrained nonconvex optimization problems where the objective function is given by the difference of two convex function (DC function). To this end, we consider a linearized proximal method with a proximal distance as regularization. Convergence analysis of particular choices of the proximal distance as second-order homogeneous proximal distances and Bregman distances are considered. Finally, some academic numerical results are presented for a constrained DC problem and generalized Fermat–Weber location problems.     

Proximity Function Minimization Using Multiple Bregman Projections, with Applications to Split Feasibility and Kullback–Leibler Distance Minimization

Problems in signal detection and image recovery can sometimes be formulated as a convex feasibility problem (CFP) of finding a vector in the intersection of a finite family of closed convex sets. Algorithms for this purpose typically employ orthogonal or generalized projections onto the individual convex sets. The simultaneous multiprojection algorithm of Censor and Elfving for solving the CFP, in which different generalized projections may be used at the same time, has been shown to converge for the case of nonempty intersection; still open is the question of its convergence when the intersection of the closed convex sets is empty.Motivated by the geometric alternating minimization approach of Csiszár and Tusnády and the product space formulation of Pierra, we derive a new simultaneous multiprojection algorithm that employs generalized projections of Bregman to solve the convex feasibility problem or, in the inconsistent case, to minimize a proximity function that measures the average distance from a point to all convex sets. We assume that the Bregman distances involved are jointly convex, so that the proximity function itself is convex. When the intersection of the convex sets is empty, but the closure of the proximity function has a unique global minimizer, the sequence of iterates converges to this unique minimizer. Special cases of this algorithm include the Expectation Maximization Maximum Likelihood (EMML) method in emission tomography and a new convergence result for an algorithm that solves the split feasibility problem.     

On inexact generalized proximal methods with a weakened error tolerance criterion

Two inexact versions of a Bregman-function-based proximal method for finding a zero of a maximal monotone operator, suggested in [J. Eckstein (1998). Approximate iterations in Bregman-function-based proximal algorithms. Math. Programming, 83, 113–123; P. da Silva, J. Eckstein and C. Humes (2001). Rescaling and stepsize selection in proximal methods using separable generalized distances. SIAM J. Optim., 12, 238–261], are considered. For a wide class of Bregman functions, including the standard entropy kernel and all strongly convex Bregman functions, convergence of these methods is proved under an essentially weaker accuracy condition on the iterates than in the original papers.

Also the error criterion of a logarithmic–quadratic proximal method, developed in [A. Auslender, M. Teboulle and S. Ben-Tiba (1999). A logarithmic-quadratic proximal method for variational inequalities. Computational Optimization and Applications, 12, 31–40], is relaxed, and convergence results for the inexact version of the proximal method with entropy-like distance functions are described.

For the methods mentioned, like in [R.T. Rockafellar (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optim., 14, 877–898] for the classical proximal point algorithm, only summability of the sequence of error vector norms is required.     

Steered sequential projections for the inconsistent convex feasibility problem

We study a steered sequential gradient algorithm which minimizes the sum of convex functions by proceeding cyclically in the directions of the negative gradients of the functions and using steered step-sizes. This algorithm is applied to the convex feasibility problem by minimizing a proximity function which measures the sum of the Bregman distances to the members of the family of convex sets. The resulting algorithm is a new steered sequential Bregman projection method which generates sequences that converge if they are bounded, regardless of whether the convex feasibility problem is or is not consistent. For orthogonal projections and affine sets the boundedness condition is always fulfilled.     

Equilibrium programming using proximal-like algorithms

We compute constrained equilibria satisfying an optimality condition. Important examples include convex programming, saddle problems, noncooperative games, and variational inequalities. Under a monotonicity hypothesis we show that equilibrium solutions can be found via iterative convex minimization. In the main algorithm each stage of computation requires two proximal steps, possibly using Bregman functions. One step serves to predict the next point; the other helps to correct the new prediction. To enhance practical applicability we tolerate numerical errors. Research supported partly by the Norwegian Research Council, project: Quantec 111039/401.     

An entropy-like proximal algorithm and the exponential multiplier method for convex symmetric cone programming

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to symmetric cone constraints. The algorithm is based on a distance-like function that is an extension of the Kullback-Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithms for convex programming with nonnegative orthant cone constraints, we show that, under some mild assumptions, the sequence generated by the proposed algorithm is bounded and every accumulation point is a solution of the considered problem. In addition, we also present a dual application of the proposed algorithm to the symmetric cone linear program, leading to a multiplier method which is shown to possess similar properties as the exponential multiplier method (Tseng and Bertsekas in Math. Program. 60:1–19, 1993) holds.     

Bregman distances and Chebyshev sets

A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given.     

On Bregman-Type Distances for Convex Functions and Maximally Monotone Operators

Given two point to set operators, one of which is maximally monotone, we introduce a new distance in their graphs. This new concept reduces to the classical Bregman distance when both operators are the gradient of a convex function. We study the properties of this new distance and establish its continuity properties. We derive its formula for some particular cases, including the case in which both operators are linear monotone and continuous. We also characterize all bi-functions D for which there exists a convex function h such that D is the Bregman distance induced by h.     

Proximal-Like Algorithm Using the Quasi D-Function for Convex Second-Order Cone Programming

In this paper, we present a measure of distance in a second-order cone based on a class of continuously differentiable strictly convex functions on ℝ₊₊. Since the distance function has some favorable properties similar to those of the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451–464 [1992]), we refer to it as a quasi D-function. Then, a proximal-like algorithm using the quasi D-function is proposed and applied to the second-cone programming problem, which is to minimize a closed proper convex function with general second-order cone constraints. Like the proximal point algorithm using the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451–464 [1992]; Chen and Teboulle in SIAM J. Optim. 3:538–543 [1993]), under some mild assumptions we establish the global convergence of the algorithm expressed in terms of function values; we show that the sequence generated by the proposed algorithm is bounded and that every accumulation point is a solution to the considered problem. Research of Shaohua Pan was partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province. Jein-Shan Chen is a member of the Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work was partially supported by National Science Council of Taiwan.     

A fast algorithm for image segmentation based on fuzzy region competition

Variational models for image segmentation are usually solved by the level set method, which is not only slow to compute but also dependent on initialization strongly. Recently, fuzzy region competition models or globally convex segmentation models have been introduced. They are insensitive to initialization, but contain TV-regularizers, making them difficult to compute. Goldstein, Bresson and Osher have applied the split Bregman iteration to globally convex segmentation models which avoided the regularization of TV norm and speeded up the computation. However, the split Bregman method needs to solve a partial differential equation (PDE) in each iteration. In this paper, we present a simple algorithm without solving the PDEs proposed originally by Jia et al. (2009) with application to image segmentation problems. The algorithm also avoids the regularization of TV norm and has a simpler form, which is in favor of implementing. Numerical experiments show that our algorithm works faster and more efficiently than other fast schemes, such as duality based methods and the split Bregman scheme.     

Convergence rate of a proximal multiplier algorithm for separable convex minimization

In this paper, we analyse the convergence rate of the proximal algorithm proposed by us in the article [A proximal multiplier method for separable convex minimization. Optimization. 2016; 65:501–537], which has been proposed to solve a separable convex minimization problem. We prove that, under mild assumptions, the primal-dual sequences of the algorithm converge linearly to the optimal solution for a class of proximal distances.     

Two-Phase Image Segmentation by Nonconvex Nonsmooth Models with Convergent Alternating Minimization Algorithms

Two-phase image segmentation is a fundamental task to partition an image into foreground and background. In this paper, two types of nonconvex and nonsmooth regularization models are proposed for basic two-phase segmentation. They extend the convex regularization on the characteristic function on the image domain to the nonconvex case, which are able to better obtain piecewise constant regions with neat boundaries. By analyzing the proposed non-Lipschitz model, we combine the proximal alternating minimization framework with support shrinkage and linearization strategies to design our algorithm. This leads to two alternating strongly convex subproblems which can be easily solved. Similarly, we present an algorithm without support shrinkage operation for the nonconvex Lipschitz case. Using the Kurdyka-Łojasiewicz property of the objective function, we prove that the limit point of the generated sequence is a critical point of the original nonconvex nonsmooth problem. Numerical experiments and comparisons illustrate the effectiveness of our method in two-phase image segmentation.     

A parameterized proximal point algorithm for separable convex optimization

In this paper, we develop a parameterized proximal point algorithm (P-PPA) for solving a class of separable convex programming problems subject to linear and convex constraints. The proposed algorithm is provable to be globally convergent with a worst-case O(1 / t) convergence rate, where t denotes the iteration number. By properly choosing the algorithm parameters, numerical experiments on solving a sparse optimization problem arising from statistical learning show that our P-PPA could perform significantly better than other state-of-the-art methods, such as the alternating direction method of multipliers and the relaxed proximal point algorithm.     

An Accelerated Inexact Proximal Point Algorithm for Convex Minimization

The proximal point algorithm is classical and popular in the community of optimization. In practice, inexact proximal point algorithms which solve the involved proximal subproblems approximately subject to certain inexact criteria are truly implementable. In this paper, we first propose an inexact proximal point algorithm with a new inexact criterion for solving convex minimization, and show its O(1/k) iteration-complexity. Then we show that this inexact proximal point algorithm is eligible for being accelerated by some influential acceleration schemes proposed by Nesterov. Accordingly, an accelerated inexact proximal point algorithm with an iteration-complexity of O(1/k ²) is proposed.     

A majorization–minimization algorithm for split feasibility problems

The classical multi-set split feasibility problem seeks a point in the intersection of finitely many closed convex domain constraints, whose image under a linear mapping also lies in the intersection of finitely many closed convex range constraints. Split feasibility generalizes important inverse problems including convex feasibility, linear complementarity, and regression with constraint sets. When a feasible point does not exist, solution methods that proceed by minimizing a proximity function can be used to obtain optimal approximate solutions to the problem. We present an extension of the proximity function approach that generalizes the linear split feasibility problem to allow for non-linear mappings. Our algorithm is based on the principle of majorization–minimization, is amenable to quasi-Newton acceleration, and comes complete with convergence guarantees under mild assumptions. Furthermore, we show that the Euclidean norm appearing in the proximity function of the non-linear split feasibility problem can be replaced by arbitrary Bregman divergences. We explore several examples illustrating the merits of non-linear formulations over the linear case, with a focus on optimization for intensity-modulated radiation therapy.     

Dykstras algorithm with bregman projections: A convergence proof

Dykstra’s algorithm and the method of cyclic Bregman projections are often employed to solve best approximation and convex feasibility problems, which are fundamental in mathematics and the physical sciences. Censor and Reich very recently suggested a synthesis of these methods, Dykstra’s algorithm with Bregman projections, to tackle a non-orthogonal best approximation problem, They obtained convergence when each constraint is a halfspace. It is shown here that this new algorithm works for general closed convex constraints; this complements Censor and Reich’s result and relates to a framework by Tseng. The proof rests on Boyle and Dykstra’s original work and on strong properties of Bregman distances corresponding to Legendre functions. Special cases and observations simplifying the implementation of the algorithm are aiso discussed     

A proximal-based decomposition method for convex minimization problems

This paper presents a decomposition method for solving convex minimization problems. At each iteration, the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables. We derive this algorithm from Rockafellar's proximal method of multipliers, which involves an augmented Lagrangian with an additional quadratic proximal term. The algorithm preserves the good features of the proximal method of multipliers, with the additional advantage that it leads to a decoupling of the constraints, and is thus suitable for parallel implementation. We allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem's data, the method is globally convergent and at a linear rate. The method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets.Corresponding author. Partially supported by Air Force Office of Scientific Research Grant 91-0008 and National Science Foundation Grant DMS-9201297.     

A proximal method with separable Bregman distances for quasiconvex minimization over the nonnegative orthant

We present an interior proximal method with Bregman distance, for solving the minimization problem with quasiconvex objective function under nonnegative constraints. The Bregman function is considered separable and zone coercive, and the zone is the interior of the positive orthant. Under the assumption that the solution set is nonempty and the objective function is continuously differentiable, we establish the well definedness of the sequence generated by our algorithm and obtain two important convergence results, and show in the main one that the sequence converges to a solution point of the problem when the regularization parameters go to zero.