Starting from a number of motivating and abundant applications in sectional sign 2, including control of robots, eigenvalue computations, mechanical stress of materials, and statistical design, the authors describe a ...
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Starting from a number of motivating and abundant applications in sectional sign 2, 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 sectional sign sectional sign 3-5, first- and second-order optimality conditions are derived for general non-linear 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 sectional sign 6. 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.
In this paper, we implement an extended version of the inexact approach proposed by Fang and Wu (1994) for solving linear semi-infinite programming problems. Some interesting numerical results are reported. The result...
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In this paper, we implement an extended version of the inexact approach proposed by Fang and Wu (1994) for solving linear semi-infinite programming problems. Some interesting numerical results are reported. The results confirm that the inexact approach is indeed more efficient and more robust than the exact approach.
In order to study the behavior of interior-point methods on very large-scale linear programming problems, we consider the application of such methods to continuous semi-infinite linear programming problems in both pri...
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In order to study the behavior of interior-point methods on very large-scale linear programming problems, we consider the application of such methods to continuous semi-infinite linear programming problems in both primal and dual form. By considering different discretizations of such problems we are led to a certain invariance property for (finite-dimensional) interior-point methods. We find that while many methods are invariant, several, including all those with the currently best complexity bound, are not. We then devise natural extensions of invariant methods to the semi-infinite case. Our motivation comes from our belief that for a method to work well on large-scale linear programming problems, it should be effective on fine discretizations of a semi-infinite problem and it should have a natural extension to the limiting semi-infinite case.
In this paper, directional differentiability properties of the optimal value function of a parameterized semi-infinite programming problem are studied. It is shown that if the unperturbed semi-infinite programming pro...
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In this paper, directional differentiability properties of the optimal value function of a parameterized semi-infinite programming problem are studied. It is shown that if the unperturbed semi-infinite programming problem is convex, then the corresponding optimal value function is directionally differentiable under mild regularity assumptions. A max-min formula for the directional derivatives, well-known in the finite convex case, is given.
The paper studies the linear semi-infinite programming problems and its dual problems. The main purpose is to develop an applicable algorithm to solve such kinds of problems.
The paper studies the linear semi-infinite programming problems and its dual problems. The main purpose is to develop an applicable algorithm to solve such kinds of problems.
In this article, we utilize the semiinfinite versions of Guignard's constraint qualification and Motzkin's theorem of the alternative to establish a set of Karush-Kuhn-Tucker-type necessary optimality conditio...
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In this article, we utilize the semiinfinite versions of Guignard's constraint qualification and Motzkin's theorem of the alternative to establish a set of Karush-Kuhn-Tucker-type necessary optimality conditions for a nonsmooth and nonconvex semiinfinite programming problem. Furthermore, we discuss some sufficient optimality conditions and duality relations for our semiinfinite programming problem.
Load duration curves play an important role in the planning practice of electric power systems. In the paper, we consider the problem of approximating a load duration curve by a polynomial under monotonicity and some ...
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Load duration curves play an important role in the planning practice of electric power systems. In the paper, we consider the problem of approximating a load duration curve by a polynomial under monotonicity and some other constraints. We show that semi-infinite programming techniques can be applied for solving this problem. A convergent inner-outer method and a finite epsilon-optimal algorithm is proposed.
In this paper we begin with pointing out how a semi-infinite programming problem can be reduced locally to a problem of finite dimensional programming. Such a reduction has the advantage that efficient numerical metho...
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In this paper we begin with pointing out how a semi-infinite programming problem can be reduced locally to a problem of finite dimensional programming. Such a reduction has the advantage that efficient numerical methods like sequential quadratic programming (SQP) methods can be applied. However, the reduced problem involves constraint functions that are defined only implicitly. Values of these functions and their derivatives must be computed iteratively with controllable errors. We interpret them as perturbations of the correct constraints and apply an SQP method with a Broyden-Fletcher-Goldfarb-Shanno (BFGS) update. Extending the convergence analysis by Fontecilla, Steihaug, and Tapia for these methods to include perturbations of the constraints and their derivatives, we are able to show q-superlinear convergence and at the same time to indicate at which rate the error in the calculation of the constraints must be reduced as the iteration progresses.
This paper presents an exhaustive approach to optimality theory in semi-infinite linear programming, placing a special emphasis on generality. After surveying optimality conditions for general problems, a detailed ana...
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This paper presents an exhaustive approach to optimality theory in semi-infinite linear programming, placing a special emphasis on generality. After surveying optimality conditions for general problems, a detailed analysis is made of problems in which the coefficients are continuous functions of a parameter which varies on a compact set, adopting a feasible directions approach. Lastly, the case of analytical coefficients over an interval is considered in some detail.
In this paper, we establish a set of necessary optimality conditions, and formulate and discuss a fairly large number of sets of global sufficient optimality criteria under various generalized (F, beta, phi, rho, thet...
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In this paper, we establish a set of necessary optimality conditions, and formulate and discuss a fairly large number of sets of global sufficient optimality criteria under various generalized (F, beta, phi, rho, theta) - univexity assumptions for a semiinfinite fractional programming problem.
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