For nonconvex problems, the saddle point equivalence of the Lagrangian approach need not hold. The nonexistence of a saddle point causes the generation of a dual gap at the solution point, and the Lagrangian approach ...
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For nonconvex problems, the saddle point equivalence of the Lagrangian approach need not hold. The nonexistence of a saddle point causes the generation of a dual gap at the solution point, and the Lagrangian approach then fails to give the solution to the original problem. Unfortunately, dual gaps are a fairly common phenomenon for engineering system design problems. Methods which are available to resolve the dual gaps destroy the separability of separable systems. The present work employs the method of multipliers by Hestenes to resolve the dual gaps of engineering system design problems; it then develops an algorithmic procedure which preserves the separability characteristics of the system. The theoretical foundations of the proposed algorithm are developed, and examples are provided to clarify the approach taken.
This paper considers the problem of choosingn numbers in the unit interval in such a way that their products in pairs are distributed as evenly as possible. Specifically, it is desired (i) to maximize the minimum diff...
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This paper considers the problem of choosingn numbers in the unit interval in such a way that their products in pairs are distributed as evenly as possible. Specifically, it is desired (i) to maximize the minimum difference between successive products or (ii) to minimize the maximum difference between successive products. These problems are solved for the casesn=1, 2, 3. For generaln, the problems are solved under certain additional restrictions, and the limiting behavior for largen is determined. The situation for generaln is investigated further by posing and solving a continuous analogue of the discrete problem. This leads to a heuristic method of determining the optimum order of products in the general case.
A nonconvex programming problem, which arises in the context of application of Benders' decomposition procedure to a class of network optimization problems, is considered. Conditions which are both necessary and s...
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A nonconvex programming problem, which arises in the context of application of Benders' decomposition procedure to a class of network optimization problems, is considered. Conditions which are both necessary and sufficient for a local maximum are derived. The concept of a basic local maximum is introduced, and it is shown that there is a finite number of basic local maxima and at least one such local maximum is optimal.
As is well known, a saddle point for the Lagrangian function, if it exists, provides a solution to a convex programming problem; then, the values of the optimal primal and dual objective functions are equal. However, ...
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As is well known, a saddle point for the Lagrangian function, if it exists, provides a solution to a convex programming problem; then, the values of the optimal primal and dual objective functions are equal. However, these results are not valid for nonconvex problems. In this paper, several results are presented on the theory of the generalized Lagrangian function, extended from the classical Lagrangian and the generalized duality program. Theoretical results for convex problems also hold for nonconvex problems by extension of the Lagrangian function. The concept of supporting hypersurfaces is useful to add a geometric interpretation to computational algorithms. This provides a basis to develop a new algorithm.
The problem of optimally allocating a fixed budget to the various arcs of a single-source, single-sink network for the purpose of maximizing network flow capacity is considered. The initial vector of arc capacities is...
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This paper considers some programming problems with absolute-value (objective) functions subject to linear constraints. Necessary and sufficient conditions for the existence of finite optimum solutions to these proble...
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This paper gives several sets of sufficient conditions that a local solution. x(k) exists of the problem min(Rk)f(k)(x), k = 1, 2, ..., such that {x(k)} has duster points that are local solutions of a problem of. the ...
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This paper gives several sets of sufficient conditions that a local solution. x(k) exists of the problem min(Rk)f(k)(x), k = 1, 2, ..., such that {x(k)} has duster points that are local solutions of a problem of. the form min(Rf)(x). The results are based on a well-known concept of topological, or point-wise convergence of the sets {R(k)} to R. Such results have been used to construct and validate large classes of mathematical programming methods based on successive approximations of the problem functions. They are also directly applicable to parametric and sensitivity analysis and provide additional characterizations of optimality for large classes of nonlinear programming problems.
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