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Projection methods for non-linear programming

Mathematical programming 1973年第1期4卷 245-268页

作者： Sargent, R.W.H. Murtagh, B.A. Imperial College London United Kingdom

Several algorithms are presented for solving the non-linear programming problem, based on "variable-metric" projections of the gradient of the objective function into a local approximation to the constraints. The algorithms differ in the nature of this approximation. Inequality constraints are dealt with by selecting at each step a subset of "active" constraints to treat as equalities, this subset being the smallest necessary to ensure that the new point remains feasible. Some numerical results are given for the Colville problems. © 1973 The Mathematical programming Society.

关键词： Objective Function Mathematical Method programming problem Projection Method Inequality Constraint

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An exact penalty function for nonlinear programming with inequalities

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Mathematical programming 1973年第1期5卷 129-150页

作者： Fletcher, R. A.E.R.E. Harwell Berkshire United Kingdom

It is shown how, given a nonlinear programming problem with inequality constraints, it is possible to construct an exact penalty function with a local unconstrained minimum at any local minimum of the constrained problem. The unconstrained minimum is sufficiently smooth to permit conventional optimization techniques to be used to locate it. Numerical evidence is presented on five well-known test problems. © 1973 The Mathematical programming Society.

关键词： Local Minimum Mathematical Method programming problem Test problem Optimization Technique

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Penalty methods for mathematical programming in Eⁿwith general constraint sets

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Journal of Optimization Theory and Applications 1970年第3期6卷 252-268页

作者： Fiacco, Anthony V. Advanced Research Department Research Analysis Corporation McLean Virginia United States

This paper extends the principal supporting results and the general convergence theorems for penalty methods, obtained by Fiacco and McCormick (Ref. 1) for the continuous mathematical programming problem, to the problem of minimizing a mildly regulated objective function over any nonempty subset of Eⁿ.The constraint set need not be defined through a collection of inequalities. A general auxiliary function is defined, and the desired minimizing sequence is shown to exist, without additional assumptions (i.e., assumptions other than those invoked in the principal convergence theorem of Ref. 1). A particularly interesting consequence is the fact that a discrete (e.g., integer) programming problem can be solved by a single unconstrained minimization of the auxiliary function. © 1970 Plenum Publishing Corporation.

关键词： programming problem Mathematical programming Convergence Theorem Additional Assumption Nonempty Subset

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On the foundations of linear and integer linear programming I

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Mathematical programming 1975年第1期9卷 207-226页

作者： Graver, Jack E. Syracuse University Syracuse New York United States

In this paper we consider the question: how does the flow algorithm and the simplex algorithm work? The usual answer has two parts: first a description of the "improvement process", and second a proof that if no further improvement can be made by this process, an optimal vector has been found. This second part is usually based on duality -a technique not available in the case of an arbitrary integer programming problem. We wish to give a general description of "improvement processes" which will include both the simplex and flow algorithms, which will be applicable to arbitrary integer programming problems, and which will in themselves assure convergence to a solution. Geometrically both the simplex algorithm and the flow algorithm may be described as follows. At the ith stage, we have a vertex (or feasible flow) to which is associated a finite set of vectors, namely the set of edges leaving that vertex (or the set of unsaturated paths). The algorithm proceeds by searching among this special set for a vector along which the gain function is increasing. If such a vector is found, the algorithm continues by moving along this vector as far as is possible while still remaining feasible. The search is then repeated at this new feasible point. We give a precise definition for sets of vectors, called test sets, which will include those sets described above arising in the simplex and flow algorithms. We will then prove that any "improvement process" which searches through a test set at each stage converges to an optimal point in a finite number of steps. We also construct specific test sets which are the natural extensions of the test sets employed by the flow algorithm to arbitrary linear and integer linear programming problems. © 1975 The Mathematical programming Society.

关键词： programming problem Integer Linear programming Natural Extension Linear programming problem Optimal Point

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Theorems on the decomposition of a large nonlinear convex separable economic system in the dual direction

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Economics of Planning 1978年第1期14卷 45-53页

作者： Sandblom, Carl-Louis Concordia University Montreal Canada

Large mathematical programming problems often arise as the result of the economic planning process. When such a problem is not only large, but nonlinear as well, there is a need to make it more manageable by breaking it down into several smaller and more easily handled subproblems. The subproblems are solved separately with the coordination activity carried out by a "master problem". A decomposition method can be seen as a "dialogue" between the master problem and the subproblems, where the flow of information back and forth between the former and the latters results in a series of approximations converging to the solution of the overall problem. Such a decomposition method was elaborated by Benders [1] for linear programmes and generalized to nonlinear convex separable programmes by Kronsjö [4] and by Geoffrion [3]. After considering our basic nonlinear programming problem from a two-stage minimization point of view, we review the Kronsjö nonlinear decomposition algorithm. Then we establish some properties of a function related to this algorithm. © 1978 The Centre for Russian and East European Studies.

关键词： Economic Growth programming problem Planning Process Mathematical programming Economic System

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Investigation of some branch and bound strategies for the solution of mixed integer linear programs

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Mathematical programming 1973年第1期4卷 155-170页

作者： Mitra, G. Brunel University Uxbridge Middlesex United Kingdom UNICOM Consultants London United Kingdom

The branch and bound method of solving the mixed integer linear programming problem is summarized. The flexibility of this technique is examined through experiments with different branching and subproblem selection st... 详细信息

关键词： Mathematical Method programming problem Selection Strategy Mixed Integer Integer Linear Program

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Generalized Benders decomposition

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Journal of Optimization Theory and Applications 1972年第4期10卷 237-260页

作者： Geoffrion, A.M. University of California at Los Angeles Los Angeles California United States

J. F. Benders devised a clever approach for exploiting the structure of mathematical programming problems with complicating variables (variables which, when temporarily fixed, render the remaining optimization problem considerably more tractable). For the class of problems specifically considered by Benders, fixing the values of the complicating variables reduces the given problem to an ordinary linear program, parameterized, of course, by the value of the complicating variables vector. The algorithm he proposed for finding the optimal value of this vector employs a cutting-plane approach for building up adequate representations of (i) the extremal value of the linear program as a function of the parameterizing vector and (ii) the set of values of the parameterizing vector for which the linear program is feasible. Linear programming duality theory was employed to derive the natural families of cuts characterizing these representations, and the parameterized linear program itself is used to generate what are usually deepest cuts for building up the representations. In this paper, Benders' approach is generalized to a broader class of programs in which the parametrized subproblem need no longer be a linear program. Nonlinear convex duality theory is employed to derive the natural families of cuts corresponding to those in Benders' case. The conditions under which such a generalization is possible and appropriate are examined in detail. An illustrative specialization is made to the variable factor programming problem introduced by R. Wilson, where it offers an especially attractive approach. Preliminary computational experience is given. © 1972 Plenum Publishing Corporation.

关键词： programming problem Duality Theory Parameterized Linear Bender Decomposition Mathematical programming problem

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A dual decomposition algorithm

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Mathematical programming 1976年第1期11卷 177-193页

作者： Lawrence Jr., John A. University of California Irvine Ca. United States California State University Fullerton Ca. United States

Suppose that a large-scale block-diagonal linear programming problem has been solved by the Dantzig-Wolfe decomposition algorithm and that an optimal solution has been attained. Suppose further that it is desired to perform a post-optimality analysis or a complete parametric analysis on the cost-coefficients or the RHS of the linking constraints. Efficient techniques for performing these analyses for the ordinary simplex case have not been easily applied to this case as one operation involves doing a minimizing ratio between all columns of two rows of the tableau. As the columns are not readily known in Dantzig-Wolfe decomposition, other techniques must be used. To date, suggested methods involve solving small linear programs to find these minimizing ratios. In this paper a method is presented which requires solving no linear programs (except possibly in the case of degeneracy of a subproblem) using and utilizing only the information typically stored for Dantzig-Wolfe decomposition. © 1976 The Mathematical programming Society.

关键词： Mathematical Method programming problem Parametric Analysis Efficient Technique Linear programming problem

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Indefinite cubic programming with standard errors in objective function

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Unternehmensforschung Operations Research - Recherche Opérationnelle 1968年第1期12卷 113-120页

作者： Bector, C.R. Department of Mathematics Indian Institute of Technology Kanpur India

This paper concerns with the problem of indefinite cubic programming in which the objective function is the product of two factors, one of which is quadratic and contains the terms with standard errors, the other being a linear factor and the constraints being linear. It has been shown that the solution of such a programming problem can be obtained by solving a convex programming problem. The problem has been extended to the case where both the factors in the objective function are quadratic factors containing terms with standard errors. Some already known results have also been deduced as particular cases of the problem discussed. © 1968 Physica-Verlag.

关键词： Objective Function Standard Error programming problem Convex programming Linear Factor

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Enumerative inequalities in integer programming

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Mathematical programming 1972年第1期2卷 32-64页

作者： Burdet, Claude-Alain Carnegie-Mellon University Pittsburgh Pennsylvinia United States

For a linear integer programming problem, the local information contained at an optimal solution {Mathematical expression} of the continuous linear programming extension stems from the theory of L.P. solutions. This paper proposes the use of environmental information (of a global nature but pertaining to the discrete vicinity of {Mathematical expression}), in order to isolate the set of integer solutions which may be considered as true candidates for the optimum. The concept of enumerative inequalities is introduced and it is shown how it can be obtained in the context of the convex outer-domain theory of Balas, Young, et al. Generally speaking, enumerative inequalities can be made arbitrarily strong (deep), but at the cost of an increasing amount of work (i.e. enumeration) for their construction. In particular cases, however, very little global information can produce enumerative inequalities stronger than any valid cut. © 1972 North-Holland Publishing Company.

关键词： Mathematical Method programming problem Integer programming Linear Integer programming Balas

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