In this paper we establish a theoretical basis for utilizing a penalty-function method to estimate sensitivity information (i.e., the partial derivatives) of a localsolution and its associated Lagrange multipliers of ...
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In this paper we establish a theoretical basis for utilizing a penalty-function method to estimate sensitivity information (i.e., the partial derivatives) of a localsolution and its associated Lagrange multipliers of a large class of nonlinear programming problems with respect to a general parametric variation in the problem functions. The local solution is assumed to satisfy the second order sufficient conditions for a strict minimum. Although theoretically valid for higher order derivatives, the analysis concentrates on the estimation of the first order (first partial derivative) sensitivity information, which can be explicitly expressed in terms of the problem functions. For greater clarity, the results are given in terms of the mixed logarithmic-barrier quadratic-loss function. However, the approach is clearly applicable toany algorithm that generates a once differentiable “solution trajectory”.
This paper describes a proposed special algorithm for a certain class of partionable linear programming problems. The matrix defining such a linear programming problem consists of a number of non-zero diagonal blocks,...
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This paper describes a proposed special algorithm for a certain class of partionable linear programming problems. The matrix defining such a linear programming problem consists of a number of non-zero diagonal blocks, plus some entirely filled rows and dito columns.
We discuss the problem of selecting the points in an (n×n) Vandermonde matrix so as to minimize the condition number of the matrix. We give numerical answers for 2≦n≦6 in the case of symmetric point configurati...
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We discuss the problem of selecting the points in an (n×n) Vandermonde matrix so as to minimize the condition number of the matrix. We give numerical answers for 2≦n≦6 in the case of symmetric point configurations. We also consider points on the non-negative real line, and give numerical results forn=2 andn=3. For generaln, the problem can be formulated as a nonlinear minimax problem with nonlinear constraints or, equivalently, as a nonlinear programming problem.
It is shown that the linear complementarity problem of finding az inRn such thatMz + q ⩾ 0, z ⩾ 0 andzT(Mz + q) = 0 can be solved by a single linear program in some important special cases including those whenM or its...
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It is shown that the linear complementarity problem of finding az inRn such thatMz + q ⩾ 0, z ⩾ 0 andzT(Mz + q) = 0 can be solved by a single linear program in some important special cases including those whenM or its inverse is a Z-matrix, that is a real square matrix with nonpositive off-diagonal elements. As a consequence certain problems in mechanics, certain problems of finding the least element of a polyhedral set and certain quadratic programming problems, can each be solved by a single linear program.
In this study, we develop optimality criteria for a mathematical programming problem. The constraints are defined by an arbitrary set as welt as infinitely many equality and inequality constraints. Necessary condition...
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In this study, we develop optimality criteria for a mathematical programming problem. The constraints are defined by an arbitrary set as welt as infinitely many equality and inequality constraints. Necessary conditions of the Kuhn-Tucker type are obtained.
This paper discusses heuristic “branch and bound” methods for solving mixed integer linear programming problems. The research presented on here is the follow on to that recorded in [3].After a resumé of the con...
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This paper discusses heuristic “branch and bound” methods for solving mixed integer linear programming problems. The research presented on here is the follow on to that recorded in [3].After a resumé of the concept of pseudo-costs and estimations, new heuristic rules for generating a tree which make use of pseudo-costs and estimations are presented. Experiments have shown that models having a low percentage of integer variables behave in a radically different way from models with a high percentage of integer variables. The new heuristic rules seem to apply generally to the first type of ***, other heuristic rules are presented that are used with models having a high percentage of integer variables and with models having a special structure (models including special ordered sets.)The rules introduced here have been implemented in the IBM Mathematical programming System Extended/370. They are used to solve large mixed integer linear programming *** results that permit comparisons to be made among the different rules are provided and discussed.
Over the past few years a number of researchers in mathematical programming became very interested in the method of the Augmented Lagrangian to solve the nonlinear programming problem. The main reason being that the A...
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Over the past few years a number of researchers in mathematical programming became very interested in the method of the Augmented Lagrangian to solve the nonlinear programming problem. The main reason being that the Augmented Lagrangian approach overcomes the ill-conditioning problem and the slow convergence of the penalty methods. The purpose of this paper is to present a new method of solving the nonlinear programming problem, which has similar characteristics to the Augmented Lagrangian method. The original nonlinear programming problem is transformed into the minimization of a leastpth objective function which under certain conditions has the same optimum as the original problem. Convergence and rate of convergence of the new method is also proved. Furthermore numerical results are presented which illustrate the usefulness of the new approach to nonlinear programming.
For nonlinear programming problems which are factorable, a computable procedure for obtaining tight underestimating convex programs is presented. This is used to exclude from consideration regions where the global min...
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For nonlinear programming problems which are factorable, a computable procedure for obtaining tight underestimating convex programs is presented. This is used to exclude from consideration regions where the global minimizer cannot exist.
This paper identifies necessary and sufficient conditions for a penalty method to yield an optimal solution or a Lagrange multiplier of a convex programming problem by means of a single unconstrained minimization. The...
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This paper identifies necessary and sufficient conditions for a penalty method to yield an optimal solution or a Lagrange multiplier of a convex programming problem by means of a single unconstrained minimization. The conditions are given in terms of properties of the objective and constraint functions of the problem as well as the penalty function adopted. It is shown among other things that all linear programs with finite optimal value satisfy such conditions when the penalty function is quadratic.
The role of 0–1 programming problems having monotone or regular feasible sets was pointed out in [6]. The solution sets of covering and of knapsack problems are examples of monotone and of regular sets respectively. ...
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The role of 0–1 programming problems having monotone or regular feasible sets was pointed out in [6]. The solution sets of covering and of knapsack problems are examples of monotone and of regular sets respectively. Some connections are established between prime implicants of a monotone or a regular Boolean functionβ on the one hand, and facets of the convex hullH of the zeros ofβ on the other. In particular (Corollary 2) a necessary and sufficient condition is given for a constraint of a covering problem to be a facet of the corresponding integer polyhedron. For any prime implicantP ofβ, a nonempty familyF(P) of facets ofH is constructed. Proposition 17 gives easy-to-determine sharp upper bounds for the coefficients of these facets whenβ is regular. A special class of prime implicants is described for regular functions and it is shown that for anyP in this class,F(P) consists of one facet ofH, and this facet has 0–1 coefficients. Every nontrivial facet ofH with 0–1 coefficients is obtained from this class.
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