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Prox-regularization of the dual method of centers for generalized fractional programs

OPTIMIZATION METHODS & SOFTWARE 2019年第3期34卷 515-545页

作者： Boufi, K. Roubi, A. Univ Hassan 1 Fac Sci & Tech Settat Morocco

We present in this paper a prox-dual regularization algorithm for solving generalized fractional programming problems. The algorithm combines the dual method of centres for generalized fractional programs and the proximal point algorithm and can handle nondifferentiable convex problems with possibly unbounded feasible constraints set. The proposed procedure generates two sequences of dual and primal values that approximate the optimal value of the considered problem respectively from below and from above at each step. It also generates a sequence of dual solutions that converges to a solution of the dual problem, and a sequence of primal solutions whose every accumulation point is a solution of the primal problem. For a class of problems, including linear fractional programs, the algorithm converges linearly.

关键词： Generalized fractional programs dual method of centres proximal point algorithm Lagrange duality

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A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality

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JOURNAL OF GLOBAL OPTIMIZATION 2019年第1期74卷 121-160页

作者： Grad, Sorin-Mihai Wilfer, Oleg Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria Tech Univ Chemnitz Fac Math D-09107 Chemnitz Germany

We investigate via a conjugate duality approach general nonlinear minmax location problems formulated by means of an extended perturbed minimal time function, necessary and sufficient optimality conditions being delivered together with characterizations of the optimal solutions in some particular instances. A parallel splitting proximal point method is employed in order to numerically solve such problems and their duals. We present the computational results obtained in matlab on concrete examples, successfully comparing these, where possible, with earlier similar methods from the literature. Moreover, the dual employment of the proximal method turns out to deliver the optimal solution to the considered primal problem faster than the direct usage on the latter. Since our technique successfully solves location optimization problems with large data sets in high dimensions, we envision its future usage on big data problems arising in machine learning.

关键词： Gauge (Minkowski) function Minimal time function Minmax multifacility location problem Sylvester problem Apollonius problem proximal point algorithm Epigraphical projection Projection operator Machine learning

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ITERATIVE METHODS FOR PARALLEL CONVEX OPTIMIZATION WITH FIXED point CONSTRAINTS

JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS

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JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS 2019年第2期3卷 115-126页

作者： Sakurai, Kaito Jimba, Takayuki Iiduka, Hideaki Meiji Univ Dept Comp Sci Tama Ku Kawasaki Kanagawa 2148571 Japan Token Pocket Inc Setagaya Ku 3-30-6 Soshigaya Tokyo 1570072 Japan

We consider the problem of minimizing the sum of convex functions over the intersection of fixed point sets of nonexpansive mappings. Two parallel optimization methods are investigated for solving this problem. One of the two methods is based on the Krasnosel'skii-Mann fixed point algorithm, and the other one is based on the Halpern fixed point algorithm. We provide their convergence analyses under certain assumptions.

关键词： Fixed point Halpern algorithm Krasnoselskii-Mann algorithm Parallel algorithm proximal point algorithm

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A generalization of linearized alternating direction method of multipliers for solving two-block separable convex programming

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2019年 357卷 251-272页

作者： Chang, Xiaokai Liu, Sanyang Zhao, Pengjun Song, Dunjiang Xidian Univ Sch Math & Stat Xian 710071 Peoples R China Lanzhou Univ Technol Coll Sci Lanzhou 730050 Gansu Peoples R China Shangluo Univ Sch Math & Comp Applicat Shangluo 726000 Peoples R China Chinese Acad Sci Inst Sci Beijing 100190 Peoples R China Chinese Acad Sci Inst Dev Beijing 100190 Peoples R China

The linearized alternating direction method of multipliers (ADMM), with indefinite proximal regularization, has been proved to be efficient for solving separable convex optimization subject to linear constraints. In this paper, we present a generalization of linearized ADMM (G-LADMM) to solve two-block separable convex minimization model, which linearizes all the subproblems by choosing a proper positive-definite or indefinite proximal term and updates the Lagrangian multiplier twice in different ways. Furthermore, the proposed G-LADMM can be expressed as a proximal point algorithm (PPA), and all the subproblems are just to estimate the proximity operator of the function in the objective. We specify the domain of the proximal parameter and stepsizes to guarantee that G-LADMM is globally convergent. It turns out that our convergence domain of the proximal parameter and stepsizes is significantly larger than other convergence domains in the literature. The numerical experiments illustrate the improvements of the proposed G-LADMM to solve LASSO and image decomposition problems. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Alternating direction method of multipliers proximal point algorithm Separable convex programming Linearization Indefinite proximal regularization LASSO

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Progressive Decoupling of Linkages in Optimization and Variational Inequalities with Elicitable Convexity or Monotonicity

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SET-VALUED AND VARIATIONAL ANALYSIS 2019年第4期27卷 863-893页

作者： Rockafellar, R. Tyrrell Univ Washington Dept Math Box 354350 Seattle WA 98195 USA

algorithms for problem decomposition and splitting in optimization and the solving of variational inequalities have largely depended on assumptions of convexity or monotonicity. Here, a way of "eliciting" convexity or monotonicity is developed which can get around that limitation. It supports a procedure called the progressive decoupling algorithm, which is derived from the proximal point algorithm through passing to a partial inverse, localizing and rescaling. In the optimization setting, elicitability of convexity corresponds to a new and very general kind of second-order sufficient condition for a local minimum. Applications are thereby opened up to problem decomposition and splitting even in nonconvex optimization, moreover with augmented Lagrangians for subproblems assisting in the implementation.

关键词： Convex nonconvex optimization Monotone nonmonotone variational inequalities Linkage problems Progressive decoupling Progressive hedging Problem decomposition Splitting Elicitable convexity Elicitable monotonicity Variational convexity Variational second-order sufficiency proximal point algorithm Method of partial inverses proximal methods of multipliers Augmented Lagrangians 65K10 65K15

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DUALITY RESULTS AND DUAL BUNDLE METHODS BASED ON THE DUAL METHOD OF CENTERS FOR MINIMAX FRACTIONAL PROGRAMS

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SIAM JOURNAL ON OPTIMIZATION 2019年第2期29卷 1578-1602页

作者： Boufi, Karima Roubi, Ahmed Univ Hassan 1 Fac Sci & Tech Lab MISI Settat Morocco

We propose new duality results for generalized fractional programs (GFP) for a wide class of problems, not limited only to the convex case. Our approach does not use Lagrangian duality, but only an equivalent form of the GFP. We present a general approximating scheme, based on the proximal point algorithm, for solving this dual program. We take advantage of the convexity property of the dual, independently of the primal properties, to build implementable bundle methods with the support of the general scheme. However, it is well known that the principal difficulty with the duality is the evaluation of the dual function. To mitigate this difficulty, we propose bundle methods that need only approximate values and approximate subgradients of the objective dual function. We prove the convergence and the rate of convergence of these algorithms. As is the case for dual algorithms, the proposed methods generate a sequence of values that converges from below to the minimal value of the GFP, and a sequence of approximate solutions that converges to a solution of the dual problem. For certain classes of problems, the convergence is at least linear.

关键词： minimax fractional programs dual method of centers proximal point algorithm bundle methods quadratic programming

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proximal bundle methods based on approximate subgradients for solving Lagrangian duals of minimax fractional programs

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JOURNAL OF GLOBAL OPTIMIZATION 2019年第2期74卷 255-284页

作者： Boualam, H. Roubi, A. Univ Hassan 1 Fac Sci & Tech Lab MISI Settat Morocco

We are interested in this work to solve the Lagrangian dual of a generalized fractional program (GFP), which gives the minimal value of the primal problem, thanks to some duality results. With the help of a general minimax equality assumption, we give duality results under minimal assumptions. Since the associated parametric programs of the dual of GFP are always concave, we use a general approximating proximal scheme to these subproblems and construct bundle methods by means of approximate values and approximate subgradients of the dual parametric function. As for all dual algorithms, the proposed methods generate a sequence of values that converges from below to the minimal value of the GFP and a sequence of approximate solutions that converges to a solution of the Lagrangian dual of the GFP. For certain classes of problems, the convergence is at least linear.

关键词： Minimax fractional programs Lagrangian duality proximal point algorithm Bundle methods

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proximal constrained optimization approach with time penalization

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ENGINEERING OPTIMIZATION 2019年第7期51卷 1207-1228页

作者： Trejo, Kristal K. Clempner, Julio B. Poznyak, Alexander S. Natl Polytech Inst Ctr Res & Adv Studies Dept Automat Control Mexico City DF Mexico Inst Politecn Nacl Escuela Super Fis & Metemat Mexico City DF Mexico Natl Polytech Inst Sch Phys & Math Mexico City DF Mexico

This article concerns a proximal-point algorithm with time penalization. The case where the cost of moving from one position to a better one is penalized by the time taken by the agent for the decision-making is studied and the restriction employing the penalty method is incorporated. It is shown that the method converges monotonically with respect to the minimal weighted norm to a unique minimal point under mild assumptions. The gradient method is employed for solving the objective function, and its convergence is proven. The rate of convergence of the method is also estimated by computing the optimal parameters. The effectiveness of the method is illustrated by a numerical optimization example employing continuous-time Markov chains.

关键词： proximal point algorithm cost of moving time penalization convergence continuous-time Markov chains

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Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging

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MATHEMATICAL PROGRAMMING 2019年第1-2期174卷 453-471页

作者： Rockafellar, R. Tyrrell Sun, Jie Univ Washington Dept Math Box 354350 Seattle WA 98195 USA Curtin Univ Dept Math & Stat Bentley WA 6102 Australia

The concept of a stochastic variational inequality has recently been articulated in a new way that is able to cover, in particular, the optimality conditions for a multistage stochastic programming problem. One of the long-standing methods for solving such an optimization problem under convexity is the progressive hedging algorithm. That approach is demonstrated here to be applicable also to solving multistage stochastic variational inequality problems under monotonicity, thus increasing the range of applications for progressive hedging. Stochastic complementarity problems as a special case are explored numerically in a linear two-stage formulation.

关键词： Progressive hedging algorithm Stochastic variational inequality problems Stochastic complementarity problems Stochastic programming problems Maximal monotone mappings proximal point algorithm Problem decomposition

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Equilibrium problems on Riemannian manifolds with applications

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2019年第2期473卷 866-891页

作者： Wang, Xiangmei Lopez, Genaro Li, Chong Yao, Jen-Chih Guizhou Univ Coll Math & Stat Guiyang 550025 Guizhou Peoples R China Univ Seville Dept Math Anal E-41080 Seville Spain Zhejiang Univ Sch Math Sci Hangzhou 310027 Zhejiang Peoples R China China Med Univ China Med Univ Hosp Res Ctr Interneural Comp Taichung 40402 Taiwan

We study equilibrium problem on general Riemannian manifolds, focusing on existence of solutions and the convexity properties of the solution set. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on this topic in the literature. We apply our results to the mixed variational inequality and the Nash equilibrium problem. Moreover, we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem. In particular, correct proofs are provided for the results claimed in J. Math. Anal. Appl. 388 (2012) 61-77 (i.e., Theorems 3.5 and 4.9 there) regarding the existence of the mixed variational inequality and the domain of the resolvent for the equilibrium problem on Hadamard manifolds. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Riemannian manifold Equilibrium problem Variational inequality problem proximal point algorithm

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