We consider a parametrized family of nonlinear programs where the feasible region is defined by equality constraints in a ball. The global optimal value is shown to be twice continuously differentiable over an open an...
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We consider a parametrized family of nonlinear programs where the feasible region is defined by equality constraints in a ball. The global optimal value is shown to be twice continuously differentiable over an open and dense set in the perturbation parameter space.
The transportation problem with fuzzy supply values of the deliverers and with fuzzy demand values of the receivers is analysed. For the solution of the problem the technique of parametric programming is used. This ma...
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The transportation problem with fuzzy supply values of the deliverers and with fuzzy demand values of the receivers is analysed. For the solution of the problem the technique of parametric programming is used. This makes it possible to obtain not only the maximizing solution (according to the Bellman-Zadeh criterion) but also other alternatives close to the optimal solution.
An algorithm is presented which solves bounded quadratic optimization problems with n variables and one linear constraint in at most O( n ) steps. The algorithm is based on a parametric approach combined with well-kno...
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An algorithm is presented which solves bounded quadratic optimization problems with n variables and one linear constraint in at most O( n ) steps. The algorithm is based on a parametric approach combined with well-known ideas for constructing efficient algorithms. It improves an O( n log n ) algorithm which has been developed for a more restricted case of the problem.
Two examples of parametric cost programming problems—one in network programming and one in NP-hard 0-1 programming—are given; in each case, the number of breakpoints in the optimal cost curve is exponential in the s...
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Two examples of parametric cost programming problems—one in network programming and one in NP-hard 0-1 programming—are given; in each case, the number of breakpoints in the optimal cost curve is exponential in the square root of the number of variables in the problem.
In this paper, we consider a general family of nonconvex programming problems. All of the objective functions of the problems in this family are identical, but their feasibility regions depend upon a parameter ϑ. This...
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In this paper, we consider a general family of nonconvex programming problems. All of the objective functions of the problems in this family are identical, but their feasibility regions depend upon a parameter ϑ. This family of problems is called a parametric nonconvex program (PNP). Solving (PNP) means finding an optimal solution for every program in the family. A prototype branch-and-bound algorithm is presented for solving (PNP). By modifying a prototype algorithm for solving a single nonconvex program, this algorithm solves (PNP) in one branch-and-bound search. To implement the algorithm, certain compact partitions and underestimating functions must be formed in an appropriate manner. We present an algorithm for solving a particular (PNP) which implements the prototype algorithm by forming compact partitions and underestimating functions based upon rules given by Falk and Soland. The programs in this (PNP) have the same concave objective function, but their feasibility regions are described by linear constraints with differing right-hand sides. Computational experience with this algorithm is reported for various problems.
For solving optimisation problems With a special structure the dual decomposition method may be effectively used under convexity assumptions. In the nonconvex case this method in the original form may fail. We propose...
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For solving optimisation problems With a special structure the dual decomposition method may be effectively used under convexity assumptions. In the nonconvex case this method in the original form may fail. We propose here a local dual method as a generalization of the global dual method for nonconvex problems. Instead of the original problem we have to solve two auxiliary problems. In the problem of the first level which is decomposable into a finite number of smaller subproblems we must compute s-stable stationary solutions depending on a parameter value ß generated in the second level. In the problem of the second level we search for such a value ß that the corresponding stationary solution is feasible for the original problem and hence a stationary solution of it. To realize this idea we can combine a suitable method for nonconvex nonsmooth optimisation in the second level and an imbedding procedure in the first one.
In parametnc computing some parts of a problem description are given as funcuons of one or more parameters which vary over a range of values Any fgxed values of the parameters specify a problem instance, and the goal ...
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The purpose of this paper is to give a geometrical answer to the question: do the strong second order sufficienty conditions hold at any local minimum point for almost all nonlinear programs? Our idea is to reduce the...
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The purpose of this paper is to give a geometrical answer to the question: do the strong second order sufficienty conditions hold at any local minimum point for almost all nonlinear programs? Our idea is to reduce the nonlinear programming problem to a finite family of 'well-behaved' nonlinear programs by perturbing the objective function in a linear fashion and perturbing the right-hand side of the constraints by adding a constants. Each of the 'well-behaved' nonlinear programs will consist of minimizing a Morse function on a manifold with boundary, where the Morse function has no critical points on the boundary. [ABSTRACT FROM AUTHOR]
This paper is concerned with an optimization-satisfaction problem to determine an optimal solution such that a certain objective function is minimized, subject to satisfaction conditions against uncertainties of any d...
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This paper is concerned with an optimization-satisfaction problem to determine an optimal solution such that a certain objective function is minimized, subject to satisfaction conditions against uncertainties of any disturbances or opponents' decisions. Such satisfaction conditions require that plural performance criteria are always less than specified values against any disturbances or opponents' decisions. Therefore, this problem is formulated as a minimization problem with the constraints which include max operations with respect to the disturbances or the opponents' decision variables. A new computational method is proposed in which a series of approximate problems transformed by applying a penalty function method to the max operations within the satisfaction conditions are solved by usual nonlinear programming. It is proved that a sequence of approximated solutions converges to a true optimal solution. The proposed algorithm may be useful for systems design under unknown parameters, process control under uncertainties, general approximation theory, and strategic weapons allocation problems.
作者：
MARTIN, DHDirector
National Research Institute for Mathematical Sciences CSIR Pretoria South Africa
Intimate relationships are investigated between connectedness properties of the lower level sets of a real functionf on a topological spaceX and the uniqueness of suitably defined minimizing sets forf. Two distinct th...
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Intimate relationships are investigated between connectedness properties of the lower level sets of a real functionf on a topological spaceX and the uniqueness of suitably defined minimizing sets forf. Two distinct theories are presented, the simpler one pertaining to the LE-level sets$$LE_\alpha (f) = \{ x \in X|f(x) \leqslant \alpha \} $$ and the other to the LT-level sets$$LT_\alpha (f) = \{ x \in X|f(x) \leqslant \alpha \} .$$ In each theory, a specific notion of minimizing set is defined in such a way that a functionf having connected level sets can have at most one minimizing set. That this uniqueness is not trivial, however, is shown by the converse result that, ifX is Hausdorff and the sets LEα(f) are all compact, then, in each theory,f has a unique minimizing set only if it has connected level sets. The paper concludes by showing that functions with connected LT-level sets arise naturally in parametric linear programming.
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