A Cutting Plane Algorithm for Solving Bilinear Programs

Nonconvex programs which have either a nonconvex minimand and/or a nonconvex feasible region have been considered by most mathematical programmers as a hopelessly difficult area of research. There are, however, two exceptions where considerable effort to obtain a global optimum is under way. One is integer linear programming and the other is nonconvex quadratic programming. This paper addresses itself to a special class of nonconvex quadratic program referred to as a "bilinear program" in the literature. We will propose here a cutting plane algorithm to solve this class of problems

A unified branch-and-bound and cutting plane algorithm for a class of nonconvex optimization problems : application to bilinear programming

A unifled approach to branch-and-bound and cutting plane methods for solving a certain class of nonconvex optimization problems is proposed. Based on this approach an implementable algorithm is obtained for programming problems with a bilinear objective function and jointly convex constraints

Maximization of a Convex Quadratic Function Under Linear Constraints

Since the appearance of a paper by H. Tui, maximization of convex function over a polytope has attracted much attention. In his paper, two algorithms were proposed: one cutting plane and the other enumerative. However, the numerical experiments reported on the naive cutting plane approach were discouraging enough to shift the researchers more to the direction of enumerative approaches. In this paper, we will develop a cutting plane algorithm for maximizing a convex quadratic function subject to linear constraints. The basic idea is much the same as Tui's method. It also parallels some of the recent results by E. Balas and C-A. Burdet. We will, however, use standard tools which are easier to understand and will fully exploit the special structure of the problem. The main purpose of the paper is to demonstrate that the full exploitation of special structure will enable us to generate a cut which is much deeper than Tui's cut and that the cutting plane algorithm can be used to solve a rather big problem efficiently

Adaptive Robust Optimization with Dynamic Uncertainty Sets for Multi-Period Economic Dispatch under Significant Wind

The exceptional benefits of wind power as an environmentally responsible renewable energy resource have led to an increasing penetration of wind energy in today's power systems. This trend has started to reshape the paradigms of power system operations, as dealing with uncertainty caused by the highly intermittent and uncertain wind power becomes a significant issue. Motivated by this, we present a new framework using adaptive robust optimization for the economic dispatch of power systems with high level of wind penetration. In particular, we propose an adaptive robust optimization model for multi-period economic dispatch, and introduce the concept of dynamic uncertainty sets and methods to construct such sets to model temporal and spatial correlations of uncertainty. We also develop a simulation platform which combines the proposed robust economic dispatch model with statistical prediction tools in a rolling horizon framework. We have conducted extensive computational experiments on this platform using real wind data. The results are promising and demonstrate the benefits of our approach in terms of cost and reliability over existing robust optimization models as well as recent look-ahead dispatch models.Comment: Accepted for publication at IEEE Transactions on Power System