Bulletin of the Japan Society for Industrial and Applied Mathematics

A theory of "discrete convex analysis" is developed for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, subgradients, the Fenchel min-max duality, separation theorems and the Lagrange duality framework for convex/nonconvex optimization. The technical development is based on matroid-theoretic concepts, in particular, submodular functions and exchange axioms. This paper extends our understanding of the relationship between convex functions and submodular functions investigated in the eighties by A. Frank, S. Fujishige, L. Lovasz and others, and also explores a novel duality framework in nonlinear integer programming.

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In recent years, the semidefinite program (abbreviated by SDP) has been studied intensively in the fields of combinatorial optimization, and systems and control theory. The SDP is a generalization of the linear program in the Euclidean space to the space of symmetric matrices. The extension of interior-point methods for linear programs to the SDP has largely contributed to the harmonious development in theory and practice in these fields. This article presents an introduction to the SDP, the SDP relaxation of the nonconvex quadratic program, and the primal-dual interior-point method for the SDP.

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This paper surveys recent developments in the area of optimization of large scale systems that have been done by the author's group. The discussion is restricted to the solution of continuous optimization problems. Key element of our optimization technology is a primal-dual interior point method that uses the barrier-penalty function as a merit function. The package NUOPT uses a line search method and a trust region method for solving large scale optimization problems. This software performs quite well for wide variety of problems from Hock and Schittkowski, and CUTE. A brief description of the modeling laguage SIMPLE is given. SIMPLE is used to describe complex systems easily. The language is implemented by using the technique of operator overloading. It uses fast automatic differentiation technique to calculate the first and second derivatives automatically. Three large scale problems which arise in practical area are also briefly discussed.

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The algorithm of shape and layout optimization method using homogenization method is described. The optimal shape and layout design is considered as the optimal distribution of the microstructure with rectangular and rectangular body holes. The continuous distribution of the sizes and rotation angles of the holes are taken as the design variables, which is discretized using the same mesh as finite element mesh used in the structural analysis. The homogenization method are used to compute the equivalent stiffness. As the optimization method the optimality criteria method is used for the computational efficiency, and the formulation in modified to make it possible to use the optimality criteria method. Several examples are shown for the demonstration of the method.

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