Quadratic maximization over {r_brace}-1, 1{r_brace}{sup n}
We consider the problem (P) max {l_brace}{1/2}xAx + cx : x {element_of} {l_brace}-1, 1{r_brace}{sup n}{r_brace} where A is a n {times} n symmetric matrix. This problem can be converted to the case where A is positive semidefinite so (P) is equivalent to the problem ({tilde P}) max {l_brace}{1/2}xAx + cx + x {element_of} [-1, 1]{sup n}{r_brace}. If in addition c = 0, we are concerned with the following problem which often arises in Numerical Analysis S{phi}{phi}{infinity}(I) = sup {l_brace}{phi}(x) : {phi}{infinity}(x) {<=} 1{r_brace} where {phi}(x) = (xAx){sup {1/2}} and {phi}{sub {infinity}} {center_dot} is I{sub {infinity}}-norm. For this problem we can prove the stability of the Lagrangian duality. Moreover, the expression of the dual objective function can be derived explicity as g{lambda} = -{lambda}/2 + K/{lambda} where K is a negative constant depending on {phi} and {phi}{sub {infinity}}. Problem ({tilde P}) can be converted to a minimization of a difference of two convex functions (d.c.) and solved efficiently by subgradient algorithms by Pham D. Tao for d.c. optimization. However, these algorithms do not guarantee to obtain a global solution. tHe most difficult problem appears to check the globality of a obtained solution. The simple expression of g{center_dot} enables us to control numerically this situation by drawing the graph of g. Other techniques in global optimization are actually being studied. As applications, we consider the problem of spin-glasses and the partition problem. The problem of spin-glasses consists in seeking a fundamental state which corresponds to the minimal energy of a spin-glasses model. This problem can be stated as problem (P) where A has zero diagonal elements and c = 0. Numerical experiences for large scale problems will be presented.
- OSTI ID:
- 36387
- Report Number(s):
- CONF-9408161-; TRN: 94:009753-0728
- Resource Relation:
- Conference: 15. international symposium on mathematical programming, Ann Arbor, MI (United States), 15-19 Aug 1994; Other Information: PBD: 1994; Related Information: Is Part Of Mathematical programming: State of the art 1994; Birge, J.R.; Murty, K.G. [eds.]; PB: 312 p.
- Country of Publication:
- United States
- Language:
- English
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