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Convex Analysisの二,三の進展について

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Introduction to linear and nonlinear programming

A variety of nonlinear, including semidefinite, relaxations have been developed in recent years for nonconvex optimization problems. Their potential can be realized only if they can be solved with sufficient speed and reliability. Unfortunately, state-of-the-art nonlinear programming codes are significantly slower and numerically unstable compared to linear programming software.In this paper, we facilitate the reliable use of nonlinear convex relaxations in global optimization via a polyhedral branch-and-cut approach. Our algorithm exploits convexity, either identified automatically or supplied through a suitable modeling language construct, in order to generate polyhedral cutting planes and relaxations for multivariate nonconvex problems. We prove that, if the convexity of a univariate or multivariate function is apparent by decomposing it into convex subexpressions, our relaxation constructor automatically exploits this convexity in a manner that is much superior to developing polyhedral outer approximators for the original function. The convexity of functional expressions that are composed to form nonconvex expressions is also automatically exploited.Root-node relaxations are computed for 87 problems from globallib and minlplib, and detailed computational results are presented for globally solving 26 of these problems with BARON 7.2, which implements the proposed techniques. The use of cutting planes for these problems reduces root-node relaxation gaps by up to 100% and expedites the solution process, often by several orders of magnitude.

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A polyhedral branch-and-cut approach to global optimization

We introduce several solution concepts for multicriteria optimization problems, give a characterization of approximately efficient elements and discuss a general scalarization procedure. Furthermore, we derive necessary and sufficient optimality conditions, a minimal point theorem, a vector-valued variational principle of Ekeland’s type, Lagrangean multiplier rules and duality statements. An overview on vector variational inequalities and vector equilibria is given. Moreover, we discuss the results for special classes of vector optimization problems (vector-valued location and approximation problems, multicriteria fractional programming and optimal control problems).

Theory of Vector Optimization

Spend your few moment to read a book even only few pages. Reading book is not obligation and force for everybody. When you don't want to read, you can get punishment from the publisher. Read a book becomes a choice of your different characteristics. Many people with reading habit will always be enjoyable to read, or on the contrary. For some reasons, this generalized concavity in optimization and economics tends to be the representative book in this website.

Generalized Concavity in Optimization and Economics

A vector equilibrium problem is defined as follows: given a closed convex subset K of a real topological Hausdorff vector space and a bifunction F(x, y) valued in a real ordered locally convex vector space, find x
*∈K such that 
$$F(x^* ,y) \nless 0$$

 for all y∈K. This problem generalizes the (scalar) equilibrium problem and the vector variational inequality problem. Extending very recent results for these two special cases, the paper establishes existence of solutions for the unifying model, assuming that F is either a pseudomonotone or quasimonotone bifunction.

Vector equilibrium problems with generalized monotone bifunctions

The chapters are as follows: Introduction to Convex and Quasiconvex Analysis (J.B.G.Frenk, G. Kassay) Criteria for Generalized Convexity and Generalized Monotonicity in the Differentiable Case (J.-P. Crouzeix) Continuity and Differentiability of Quasiconvex Functions (J.-P. Crouzeix) Generalized Convexity and Optimality Conditions in Scalar and Vector Optimization (A. Cambini, L. Martein) Generalized Convexity in Vector Optimization (D. T. Luc) Generalized Convex Duality and Its Economic Applications (J.-E. Martinez-Legaz) Abstract Convexity (A.Rubinov, J.Dutta) Fractional programming (J.B.G. Frenk, S.Schaible) Generalized Monotone Maps (N.Hadjisavvas, S.Schaible) Generalized Convexity and Generalized Derivatives (S.Komlosi) Generalized Convexity, Generalized Monotonicity and Nonsmooth Analysis (N.Hadjisavvas) Pseudomonotone Complementarity Problems and Variational Inequalities (J.-C. Yao, O. Chadly) Generalized Monotone Equilibrium Problems and Variational Inequalities (I. Konnov) Uses of Generalized Convexity and Monotonicity in Economics (R. John)

Handbook of Generalized Convexity and Generalized Monotonicity

An algorithm is suggested that finds the constrained minimum of the maximum of finitely many ratios. The method involves a sequence of linear (convex) subproblems if the ratios are linear (convex-concave). Convergence results as well as rate of convergence results are derived. Special consideration is given to the case of (a) compact feasible regions and (b) linear ratios.

An Algorithm for Generalized Fractional Programs

Dinkelbach's algorithm [Dinkelbach, W. 1967. On nonlinear fractional programming. Management Sci.13 492--498.] solving the parametric equivalent of a fractional program is investigated. It is shown that the algorithm converges superlinearly and often locally quadratically. A priori and a posteriori error estimates are derived. Using those estimates and duality as introduced in Part I, a revised version of the algorithm is proposed. In addition, a similar algorithm is presented where, in contrast to Dinkelbach's procedure, the rate of convergence is still controllable. Error estimates are derived also for this algorithm.

Siegfried Schaible

Papers

Generalized Concavity in Optimization and Economics

Vector equilibrium problems with generalized monotone bifunctions

Handbook of Generalized Convexity and Generalized Monotonicity

An Algorithm for Generalized Fractional Programs

Fractional Programming. II, On Dinkelbach's Algorithm