Method for minimizing a convex-concave function over a convex set

A branch-and-bound method is proposed for minimizing a convex-concave function over a convex set. The minimization of a DC-function is a special case, where the subproblems connected with the bounding operation can be solved effectively.on leave at Mannheim University by a grant from the Alexander von Humboldt Foundation.     

An algorithm to solve polyhedral convex set optimization problems

An algorithm which computes a solution of a set optimization problem is provided. The graph of the objective map is assumed to be given by finitely many linear inequalities. A solution is understood to be a set of points in the domain satisfying two conditions: the attainment of the infimum and minimality with respect to a set relation. In the first phase of the algorithm, a linear vector optimization problem, called the vectorial relaxation, is solved. The resulting pre-solution yields the attainment of the infimum but, in general, not minimality. In the second phase of the algorithm, minimality is established by solving certain linear programs in combination with vertex enumeration of some values of the objective map.     

An outer approximation method for minimizing the product of several convex functions on a convex set

This paper addresses the minimization of the product ofp convex functions on a convex set. It is shown that this nonconvex problem can be converted to a concave minimization problem withp variables, whose objective function value is determined by solving a convex minimization problem. An outer approximation method is proposed for obtaining a global minimum of the resulting problem. Computational experiments indicate that this algorithm is reasonable efficient whenp is less than 4.This research was partly supported by Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture, Grant No. (C)03832018 and (C)04832010.     

An all-linear programming relaxation algorithm for optimizing over the efficient set

The problem (P) of optimizing a linear function over the efficient set of a multiple objective linear program has many important applications in multiple criteria decision making. Since the efficient set is in general a nonconvex set, problem (P) can be classified as a global optimization problem. Perhaps due to its inherent difficulty, it appears that no precisely-delineated implementable algorithm exists for solving problem (P) globally. In this paper a relaxation algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact optimal solution to the problem after a finite number of iterations. A detailed discussion is included of how to implement the algorithm using only linear programming methods. Convergence of the algorithm is proven, and a sample problem is solved.Research supported by a grant from the College of Business Administration, University of Florida, Gainesville, Florida, U.S.A.     

Note: An algorithm to solve polyhedral convex set optimization problems

We provide a counterexample to the remark in Löhne and Schrage [An algorithm to solve polyhedral convex set optimization problems, Optimization 62 (2013), pp. 131-141] that every solution of a polyhedral convex set optimization problem is a pre-solution. A correct statement is that every solution of a polyhedral convex set optimization problem obtained by the algorithm SetOpt is a pre-solution. We also show that every finite infimizer and hence every solution of a polyhedral convex set optimization problem contains a pre-solution.     

Branch-and-reduce algorithm for convex programs with additional multiplicative constraints

This article presents a branch-and-reduce algorithm for globally solving for the first time a convex minimization problem (P) with p?1$p?1$ additional multiplicative constraints. In each of these p   additional constraints, the product of two convex functions is constrained to be less than or equal to a positive number. The algorithm works by globally solving a 2p$2p$-dimensional master problem (MP) equivalent to problem (P). During a typical stage k of the algorithm, a point is found that minimizes the objective function of problem (MP) over a nonconvex set F^k$F^k$ that contains the portion of the boundary of the feasible region of the problem where a global optimal solution lies. If this point is feasible in problem (MP), the algorithm terminates. Otherwise, the algorithm continues by branching and creating a new, reduced nonconvex set F^k+1$F^{k+1}$ that is a strict subset of F^k$F^k$. To implement the algorithm, all that is required is the ability to solve standard convex programming problems and to implement simple algebraic steps. Convergence properties of the algorithm are given, and results of some computational experiments are reported.     

An iterative method for minimizing a convex nonsmooth function on a convex smooth surface

An iterative algorithm is proposed for the constrained minimization of a convex nonsmooth function on a set given as a convex smooth surface. The convergence of the algorithm in the sense of necessary conditions for a local minimum is proved.     

A note on maximizing a submodular set function subject to a knapsack constraint

In this paper, we obtain an (1−e⁻¹)-approximation algorithm for maximizing a nondecreasing submodular set function subject to a knapsack constraint. This algorithm requires O(n⁵) function value computations.     

On the convergence rate of a forward-backward type primal-dual splitting algorithm for convex optimization problems

In this paper, we analyse the convergence rate of the sequence of objective function values of a primal-dual proximal-point algorithm recently introduced in the literature for solving a primal convex optimization problem having as objective the sum of linearly composed infimal convolutions, nonsmooth and smooth convex functions and its Fenchel-type dual one. The theoretical part is illustrated by numerical experiments in image processing.     

An algorithm for maximizing entropy subject to simple bounds

An unusual form of the maximum entropy problem is considered, that includes simple bound constraints on the Fourier coefficients of the required image, as well as nonnegativity conditions on the image intensities. The algorithm avoids mixing these constraints by introducing a parameter into the objective function that is adjusted by an outer iteration. For each parameter value an inner iteration solves a large optimization calculation, whose constraints are just the simple bounds, by a combination of the conjugate gradient procedure and an active set method. An important feature is the ability to make many changes to the active set at once. The outer iteration includes a test for inconsistency of all the given constraints. The algorithm is described, a proof of convergence is given, and there are some second-hand remarks on numerical results.     

Characterizations of set order relations and constrained set optimization problems via oriented distance function

Set-valued optimization problems are important and fascinating field of optimization theory and widely applied to image processing, viability theory, optimal control and mathematical economics. There are two types of criteria of solutions for the set-valued optimization problems: the vector criterion and the set criterion. In this paper, we adopt the set criterion to study the optimality conditions of constrained set-valued optimization problems. We first present some characterizations of various set order relations using the classical oriented distance function without involving the nonempty interior assumption on the ordered cones. Then using the characterizations of set order relations, necessary and sufficient conditions are derived for four types of optimal solutions of constrained set optimization problem with respect to the set order relations. Finally, the image space analysis is employed to study the c-optimal solution of constrained set optimization problems, and then optimality conditions and an alternative result for the constrained set optimization problem are established by the classical oriented distance function.     

The e-support function of an e-convex set and conjugacy for e-convex functions

A subset of a locally convex space is called e-convex if it is the intersection of a family of open halfspaces. An extended real-valued function on such a space is called e-convex if its epigraph is e-convex. In this paper we introduce a suitable support function for e-convex sets as well as a conjugation scheme for e-convex functions.     

Testing the ℜ- strategy for a Reverse Convex Problem

This paper is devoted to solving a reverse-convex problem. The approach presented here is based on Global Optimality Conditions. We propose a general conception of a Global Search Algorithm and develop each part of it. The results of numerical experiments with the dimension up to 400 are also given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Simplified infeasible interior-point algorithm for linear optimization based on a simple function

Based on a similar kernel function, we present an infeasible version of the interior-point algorithm for linear optimization introduced by Wang et al. (2016). The property of exponential convexity is still important to simplify the analysis of the algorithm. The iteration bound coincides with the currently best iteration bound for infeasible interior-point algorithms.     

求总极值问题的最优性条件

郑权提出了求总极值问题的积分－水平集的概念性算法,同时给出了最优性条件。本文提出了修正的积分－水平集算法,并且给出了类似的总极值存在的最优性条件。     

A set-valued Lagrange theorem based on a process for convex vector programming

ABSTRACT

In this paper, we present a new set-valued Lagrange multiplier theorem for constrained convex set-valued optimization problems. We introduce the novel concept of Lagrange process. This concept is a natural extension of the classical concept of Lagrange multiplier where the conventional notion of linear continuous operator is replaced by the concept of closed convex process, its set-valued analogue. The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.     

An elementary new proof of the determination of a convex function by its subdifferential

Rockafellar proved that any closed, convex function is uniquely determined by its subdifferential mapping up to an additive constant. The aim of this article is to provide an elementary proof of the same result.     

On several results about convex set functions

In 1979, in an interesting paper, R.J. Morris introduced the notion of convex set function defined on an atomless finite measure space. After a short period this notion, as well as generalizations of it, began to be studied in several papers. The aim was to obtain results similar to those known for usual convex (or generalized convex) functions. Unfortunately several notions are ambiguous and the arguments used in the proofs of several results are not clear or not correct. In this way there were stated even false results. The aim of this paper is to point out that using some simple ideas it is possible, on one hand, to deduce the correct results by means of convex analysis and, on the other hand, to emphasize the reasons for which there are problems with other results.     

Global minimization of a generalized convex multiplicative function

This paper discusses an algorithm for generalized convex multiplicative programming problems, a special class of nonconvex minimization problems in which the objective function is expressed as a sum ofp products of two convex functions. It is shown that this problem can be reduced to a concave minimization problem with only 2p variables. An outer approximation algorithm is proposed for solving the resulting problem.