A Strongly Convergent Proximal Point Method for Vector Optimization

Journal of Optimization Theory and Applications - In this paper, we propose and analyze a variant of the proximal point method for obtaining weakly efficient solutions of convex vector optimization...     

Unconstrained Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds

In this paper, we present a steepest descent method with Armijo??s rule for multicriteria optimization in the Riemannian context. The sequence generated by the method is guaranteed to be well defined. Under mild assumptions on the multicriteria function, we prove that each accumulation point (if any) satisfies first-order necessary conditions for Pareto optimality. Moreover, assuming quasiconvexity of the multicriteria function and nonnegative curvature of the Riemannian manifold, we prove full convergence of the sequence to a critical Pareto point.     

An Inexact Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds

In this paper, we present an inexact version of the steepest descent method with Armijo’s rule for multicriteria optimization in the Riemannian context given in Bento et al. (J. Optim. Theory Appl., 154: 88–107, 2012). Under mild assumptions on the multicriteria function, we prove that each accumulation point (if any) satisfies first-order necessary conditions for Pareto optimality. Moreover, assuming that the multicriteria function is quasi-convex and the Riemannian manifold has nonnegative curvature, we show full convergence of any sequence generated by the method to a Pareto critical point.     

Generalized Proximal Method for Efficient Solutions in Vector Optimization

This article is devoted to developing the generalized proximal algorithm of finding efficient solutions to the vector optimization problem for a mapping from a uniformly convex and uniformly smooth Banach space to a real Banach space with respect to the partial order induced by a pointed closed convex cone. In contrast to most published literature on this subject, our algorithm does not depend on the nonemptiness of ordering cone of the space under consideration and deals with finding efficient solutions of the vector optimization problem in question. We prove that under some suitable conditions the sequence generated by our method weakly converges to an efficient solution of this problem.     

A New Approach to a Multicriteria Optimization Problem

We present a new approach to a multicriteria optimization problem, where the objective and the constraints are linear functions. From an equivalent equilibrium problem, first suggested in [5,6,8], we show new characterizations of weakly efficient points based on the partial order induced by a nonempty closed convex cone in a finite-dimensional linear space, as in [7]. Thus, we are able to apply the analytic center cutting plane algorithm that finds equilibrium points approximately, by Raupp and Sosa [10], in order to find approximate weakly efficient solutions of MOP.     

Constrained Nonconvex Nonsmooth Optimization via Proximal Bundle Method

In this paper, we consider a constrained nonconvex nonsmooth optimization, in which both objective and constraint functions may not be convex or smooth. With the help of the penalty function, we transform the problem into an unconstrained one and design an algorithm in proximal bundle method in which local convexification of the penalty function is utilized to deal with it. We show that, if adding a special constraint qualification, the penalty function can be an exact one, and the sequence generated by our algorithm converges to the KKT points of the problem under a moderate assumption. Finally, some illustrative examples are given to show the good performance of our algorithm.     

A Logarithmic-Quadratic Proximal Method for Variational Inequalities

We present a new method for solving variational inequalities on polyhedra. The method is proximal based, but uses a very special logarithmic-quadratic proximal term which replaces the usual quadratic, and leads to an interior proximal type algorithm. We allow for computing the iterates approximately and prove that the resulting method is globally convergent under the sole assumption that the optimal set of the variational inequality is nonempty.     

Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

This paper studies a general vector optimization problem of finding weakly efficient points for mappings from Hilbert spaces to arbitrary Banach spaces, where the latter are partially ordered by some closed, convex, and pointed cones with nonempty interiors. To find solutions of this vector optimization problem, we introduce an auxiliary variational inequality problem for a monotone and Lipschitz continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem under consideration by combining an extragradient method to find a solution of the variational inequality problem and an approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone and Lipschitz continuous mapping, and then finding weakly efficient points for a suitable regularization of the original mapping. We present both absolute and relative versions of our hybrid algorithm in which the subproblems are solved only approximately. The weak convergence of the generated sequence to a weak efficient point is established under quite mild assumptions. In addition, we develop some extensions of our hybrid algorithms for vector optimization by using Bregman-type functions.     

Multicriteria Approach to Bilevel Optimization

In this paper, we study the relationship between bilevel optimization and multicriteria optimization. Given a bilevel optimization problem, we introduce an order relation such that the optimal solutions of the bilevel problem are the nondominated points with respect to the order relation. In the case where the lower-level problem of the bilevel optimization problem is convex and continuously differentiable in the lower-level variables, this order relation is equivalent to a second, more tractable order relation. Then, we show how to construct a (nonconvex) cone for which we can prove that the nondominated points with respect to the order relation induced by the cone are also nondominated points with respect to any of the two order relations mentioned before. We comment also on the practical and computational implications of our approach.     

A Regularization Method for the Proximal Point Algorithm

A regularization method for the proximal point algorithm of finding a zero for a maximal monotone operator in a Hilbert space is proposed. Strong convergence of this algorithm is proved.Hong-Kun Xu: Supported in part by NRF     

Symmetric Alternating Direction Method with Indefinite Proximal Regularization for Linearly Constrained Convex Optimization

The proximal alternating direction method of multipliers is a popular and useful method for linearly constrained, separable convex problems, especially for the linearized case. In the literature, convergence of the proximal alternating direction method has been established under the assumption that the proximal regularization matrix is positive semi-definite. Recently, it was shown that the regularizing proximal term in the proximal alternating direction method of multipliers does not necessarily have to be positive semi-definite, without any additional assumptions. However, it remains unknown as to whether the indefinite setting is valid for the proximal version of the symmetric alternating direction method of multipliers. In this paper, we confirm that the symmetric alternating direction method of multipliers can also be regularized with an indefinite proximal term. We theoretically prove the global convergence of the indefinite method and establish its worst-case convergence rate in an ergodic sense. In addition, the generalized alternating direction method of multipliers proposed by Eckstein and Bertsekas is a special case in our discussion. Finally, we demonstrate the performance improvements achieved when using the indefinite proximal term through experimental results.     

Scalarization Method in Multicriteria Games

Using a two-criteria two-person game as an example, the validity of the scalarization method applied for the parameterization of the set of game values and for estimating the players’ payoffs is investigated. It is shown that the use of linear scalarization by the players gives the results different from those obtained using Germeyer’s scalarization. Various formalizations of the concept of value of MC games are discussed.     

区间直觉模糊集多属性决策方法

定义了区间直觉模糊集的加权算子和加权几何集成算子,介绍了现有的区间直觉模糊集的得分函数和精确函数.定义了一个新的精确函数,此函数弥补了已有函数的不足和缺陷,应用新定义的精确函数,提出了对区间直觉模糊集多属性决策问题进行决策的方法.最后以应用实例对该方法进行说明和验证.     

A Logarithmic-Quadratic Proximal Prediction-Correction Method for Structured Monotone Variational Inequalities

Inspired by the Logarithmic-Quadratic Proximal (LQP) method for variational inequalities, we present a prediction-correction method for structured monotone variational inequalities. Each iteration of the new method consists of a prediction and a correction. Both the predictor and the corrector are obtained easily with tiny computational load. In particular, the LQP system that appears in the prediction is approximately solved under significantly relaxed inexactness restriction. Global convergence of the new method is proved under mild assumptions. In addition, we present a self-adaptive version of the new method that leads to easier implementations. Preliminary numerical experiments for traffic equilibrium problems indicate that the new method is effectively applicable in practice. Presented at the 6th International conference on Optimization: Techniques and Applications, Ballarat Australia, December 9–11, 2004. This author was supported by NSFC Grant 10571083, the MOEC grant 20020284027 and Jiangsu NSF grant BK2002075     

一种基于GP算法的多准则决策函数构造方法

提出了一种基于遗传程序设计算法(GPA)构造多准则决策函数新方法,该方法构造的决策函数比典型的分层处理AHP(算术平均值)方法构造的决策函具有明显的稳定性.理论上期望得到实例的验证.     

A Proximal Point Method in Nonreflexive Banach Spaces

We propose an inexact version of the proximal point method and study its properties in nonreflexive Banach spaces which are duals of separable Banach spaces, both for the problem of minimizing convex functions and of finding zeroes of maximal monotone operators. By using surjectivity results for enlargements of maximal monotone operators, we prove existence of the iterates in both cases. Then we recover most of the convergence properties known to hold in reflexive and smooth Banach spaces for the convex optimization problem. When dealing with zeroes of monotone operators, our convergence result requests that the regularization parameters go to zero, as is the case for standard (non-proximal) regularization schemes.     

An Adaptive Proximal Method for Variational Inequalities

Computational Mathematics and Mathematical Physics - A novel analog of Nemirovski’s proximal mirror method with an adaptive choice of constants in the minimized prox-mappings at each...     

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.     

A Proximal Bundle Method Based on Approximate Subgradients

In this paper a proximal bundle method is introduced that is capable to deal with approximate subgradients. No further knowledge of the approximation quality (like explicit knowledge or controllability of error bounds) is required for proving convergence. It is shown that every accumulation point of the sequence of iterates generated by the proposed algorithm is a well-defined approximate solution of the exact minimization problem. In the case of exact subgradients the algorithm behaves like well-established proximal bundle methods. Numerical tests emphasize the theoretical findings.