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Semi-Infinite Programming: Theory, Methods, and Applications

R. Hettich and K. O. Kortanek
SIAM Review
Vol. 35, No. 3 (Sep., 1993), pp. 380-429
Stable URL: http://www.jstor.org/stable/2132425
Page Count: 50
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Semi-Infinite Programming: Theory, Methods, and Applications
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Abstract

Starting from a number of motivating and abundant applications in $\S2$, including control of robots, eigenvalue computations, mechanical stress of materials, and statistical design, the authors describe a class of optimization problems which are referred to as semi-infinite, because their constraints bound functions of a finite number of variables on a whole region. In $\S\S3-5$, first- and second-order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints. Another main effort for achieving simplification is through duality in $\S6$. There, algebraic properties of finite linear programming are brought to bear on duality theory in semi-infinite programming. Section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent (SQP-type) methods. Taking this differentiable point of view, this paper can be considered to be complementary to the review given by Polak [SIAM Rev., 29 (1987), pp. 21-89] on the nondifferentiable approach. The last, short section briefly reviews some work done on parametric problems.

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