| Budget Amount *help |
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2006: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2005: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Research Abstract |
We mainly studied two classes of nonconvex optimization problems : concave minimization and reverse convex minimization problems. The former minimizes a concave function over a polyhedron, and the latter minimizes a linear function but the feasible set is the difference of two convex sets. Therefore, each problem has multiple locally optimal solutions, many of which fail to be globally optimal. To solve them rigorously, we usually apply branch-and-bound algorithms, which implicitly enumerate locally optimal solutions. Unfortunately, however, those algorithms have two serious drawbacks and cannot solve problems of utility-scale in a practical amount of time. First, the algorithms partition the feasible set into simplices or cones, and estimate lower bounds on each partition set by solving a linearized problem. Although real-world problems often possess favorable structures, those are destroyed if the feasible sets are partitioned. Second, the algorithms easily suffers from combinatorial
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explosion, far before yielding expected solutions.
Based on the above observations, we proposed novel branch-and-bound algorithms conscious of special structures. The main idea is simply to stop partitioning the feasible sets. This approach, however, deteriorates the lower bounds considerably. To tighten them, we devised an inexpensive procedure using a Lagrangian relaxation. We confirmed that it works well on instances of concave minimization, reverse convex minimization problems, and production-transportation problems with network structure. In dealing with combinatorial explosion, we proposed a combinatorial branch-and-bound algorithm to solve a reverse convex minimization problem whose feasible set is the difference of a polytope and a convex set. This algorithm searches locally optimal solutions along edges of the polytope using reverse Bland's pivoting operations. We proved that it visits each locally optimal solution just one time and besides needs no list of visiting locally optimal solutions. Therefore, the algorithm yields a globally optimal solution in finite time with polynomial-space complexity. Less
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