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**1 - 2**of**2**### Deterministic Algorithms for Some Global Optimization Problems

, 2006

"... We study two major classes of problems in global optimization, concave minimization and reverse convex programming. The former problem is of minimizing a concave function under linear constraints and the latter problem is of minimizing a linear function over the intersection of a convex and the comp ..."

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We study two major classes of problems in global optimization, concave minimization and reverse convex programming. The former problem is of minimizing a concave function under linear constraints and the latter problem is of minimizing a linear function over the intersection of a convex and the complement of a convex set. In general, both problems have enormous number of locally optimal solutions, among which the usual algorithms can fail to find a globally optimal one. To locate a globally optimal solution without fail in a practical amount of time, we propose branch-and-bound algorithms incorporating some procedures for accelerating convergence. The main idea behind the algorithms is to exploit special structures potentially possessed by the real-world problems, e.g., network structures and low-rank nonconvexity. We assume that each target problem has some favorable structures of these kinds and define relaxation problems needed to solve in the bounding process in such a way that they inherit the structures from the target problem. While this approach enables us to solve each relaxation problem efficiently, the resulting lower bounds on the optimal value become worse than

### A Polynomial-Space Finite Algorithm for Solving a Class of Reverse Convex Programs

, 2007

"... We develop a new kind of branch-and-bound algorithm to solve a linear program with an additional reverse convex constraint. The proposed algorithm is based on a polynomial-space pivoting algorithm for enumerating feasible bases of a linear program. We show that it generates a globally optimal soluti ..."

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We develop a new kind of branch-and-bound algorithm to solve a linear program with an additional reverse convex constraint. The proposed algorithm is based on a polynomial-space pivoting algorithm for enumerating feasible bases of a linear program. We show that it generates a globally optimal solution after finitely many pivoting operations with polynomial space.