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Convex Analysisの二,三の進展について

We consider the problem of finding the unconstrained global minimum of a real-valued polynomial p(x): {\mathbb{R}}^n\to {\mathbb{R}}$, as well as the global minimum of p(x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems. A notion of Karush--Kuhn--Tucker polynomials is introduced in a global optimality condition. Some illustrative examples are provided.

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Global Optimization with Polynomials and the Problem of Moments

Convex Analysis: (pms-28)

The DC programming and its DC algorithm (DCA) address the problem of minimizing a function f=g−h (with g,h being lower semicontinuous proper convex functions on R

 n
 ) on the whole space. Based on local optimality conditions and DC duality, DCA was successfully applied to a lot of different and various nondifferentiable nonconvex optimization problems to which it quite often gave global solutions and proved to be more robust and more efficient than related standard methods, especially in the large scale setting. The computational efficiency of DCA suggests to us a deeper and more complete study on DC programming, using the special class of DC programs (when either g or h is polyhedral convex) called polyhedral DC programs. The DC duality is investigated in an easier way, which is more convenient to the study of optimality conditions. New practical results on local optimality are presented. We emphasize regularization techniques in DC programming in order to construct suitable equivalent DC programs to nondifferentiable nonconvex optimization problems and new significant questions which have to be answered. A deeper insight into DCA is introduced which really sheds new light on DCA and could partly explain its efficiency. Finally DC models of real world nonconvex optimization are reported.

The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems

Dedicated to Hoang Tuy on the occasion of his seventieth birthday Abstract. This paper is devoted to a thorough study on convex analysis approach to d.c. (difierence of convex functions) programming and gives the State of the Art. Main results about d.c. duality, local and global opti- malities in d.c. programming are presented. These materials constitute the basis of the DCA (d.c. algorithms). Its convergence properties have been tackled in detail, especially in d.c. polyhedral programming where it has flnite convergence. Exact penalty, Lagrangian duality without gap, and regularization techniques have beeen studied to flnd appropriate d.c. de- compositions and to improve consequently the DCA. Finally we present the application of the DCA to solving a lot of important real-life d.c. pro- grams.

Convex analysis approach to d.c. programming: Theory, Algorithm and Applications

The year 2015 marks the 30th birthday of DC (Difference of Convex functions) programming and DCA (DC Algorithms) which constitute the backbone of nonconvex programming and global optimization. In this article we offer a short survey on thirty years of developments of these theoretical and algorithmic tools. The survey is comprised of three parts. In the first part we present a brief history of the field, while in the second we summarize the state-of-the-art results and recent advances. We focus on main theoretical results and DCA solvers for important classes of difficult nonconvex optimization problems, and then give an overview of real-world applications whose solution methods are based on DCA. The third part is devoted to new trends and important open issues, as well as suggestions for future developments.

DC programming and DCA: thirty years of developments

In the present paper, we are concerned with conditions ensuring the exact penalty for nonconvex programming. Firstly, we consider problems with concave objective and constraints. Secondly, we establish various results on error bounds for systems of DC inequalities and exact penalty, with/without error bounds, in DC programming. They permit to recast several class of difficult nonconvex programs into suitable DC programs to be tackled by the efficient DCA.

Exact penalty and error bounds in DC programming

Difference of Convex functions DC Programming and DC Algorithm DCA constitute the backbone of Nonconvex Programming and Global Optimization The paper is devoted to the State of the Art with recent advances of DC Programming and DCA to meet the growing need for nonconvex optimization and global optimization, both in terms of mathematical modeling as in terms of efficient scalable solution methods After a brief summary of these theoretical and algorithmic tools, we outline the main results on convergence of DCA in DC programming with subanalytic data, exact penalty techniques with/without error bounds in DC programming including mixed integer DC programming, DCA for general DC programs, and DC programming involving the l0-norm via its approximation and penalization

Recent Advances in DC Programming and DCA

Feature selection consists of choosing a subset of available features that capture the relevant properties of the data. In supervised pattern classification, a good choice of features is fundamental for building compact and accurate classifiers. In this paper, we develop an efficient feature selection method using the zero-norm l
 0 in the context of support vector machines (SVMs). Discontinuity at the origin for l
 0 makes the solution of the corresponding optimization problem difficult to solve. To overcome this drawback, we use a robust DC (difference of convex functions) programming approach which is a general framework for non-convex continuous optimisation. We consider an appropriate continuous approximation to l
 0 such that the resulting problem can be formulated as a DC program. Our DC algorithm (DCA) has a finite convergence and requires solving one linear program at each iteration. Computational experiments on standard datasets including challenging feature-selection problems of the NIPS 2003 feature selection challenge and gene selection for cancer classification show that the proposed method is promising: while it suppresses up to more than 99% of the features, it can provide a good classification. Moreover, the comparative results illustrate the superiority of the proposed approach over standard methods such as classical SVMs and feature selection concave.

Hoai An Le Thi

Papers

Convex analysis approach to d.c. programming: Theory, Algorithm and Applications

DC programming and DCA: thirty years of developments

Exact penalty and error bounds in DC programming

Recent Advances in DC Programming and DCA

A DC programming approach for feature selection in support vector machines learning