Showing papers in "Optimization in 1992"
TL;DR: The Kuhn-Tucker conditions of an optimization problem with inequality constraints are transformed equivalently into a special nonlinear system of equations T 0(z) = 0 as mentioned in this paper, and Newton's method for solving this system combines two valuable properties: the local Q-quadratic convergence without assuming the strict complementary slackness condition and the regularity of the Jacobian of T 0 at a point z under reasonable conditions.
Abstract: The Kuhn–Tucker conditions of an optimization problem with inequality constraints are transformed equivalently into a special nonlinear system of equations T 0(z) = 0. It is shown that Newton's method for solving this system combines two valuable properties: The local Q-quadratic convergence without assuming the strict complementary slackness condition and the regularity of the Jacobian of T 0 at a point z under reasonable conditions, so that Newton’s method can be used also far from a Kuhn–Tucker point
718 citations
TL;DR: In this article, the authors present a list of 491 papers dealing with fractional programming and its applications, covering mainly the period 1997-2005 but also including some references published up to 1997 which were not included in the previous bibliographies or which were mentioned as ‘to appear’ in the five previous bibliography.
Abstract: This bibliography of fractional programming is a continuation of five previous bibliographies by the author (Pure Appl. Math. Sci. (India), Vol. XIII, No. 1–2, 35–69, March (1981); ibid. Vol. XVII, No. 1–2, 87–102, March (1983); ibid. XXII, No. 1–2, 109–122, September (1985); Optimization 23(1992)1, 53–71; ibid. 45(1999) 1–4, 343–367). This compilation lists, in alphabetical order by the name of the first author, 491 papers dealing with fractional programming and its applications. This covers mainly the period 1997–2005 but it also includes some references published up to 1997 which were not included in the previous bibliographies or which were mentioned as ‘to appear’ in the five bibliography. In compiling this list we used Mathematical Reviews, Zentralblatt fur Mathematik, Referativnyi Zhurnal (Matematika) and Current Papers on Computers & Control. The papers are either published in some form (in technical journals or as internal reports) or are available only as typewritten manuscripts (for example, as...
125 citations
TL;DR: In this paper, a vector valued variational principle by using a general concept of ∊-efficiency and a nonconvex separation theorem is presented. But this principle is not applicable to the problem of vector valued VAE.
Abstract: This paper presents a vector valued variational principle by using a general concept of ∊-efficiency and a nonconvex separation theorem
110 citations
TL;DR: In this article, an algorithm for the pooling problem in refinery optimization based on a bilinear programming approach is presented, and the method is illustrated on numerical examples in numerical simulations.
Abstract: In this paper we present an algorithm for the pooling problem in refinery optimization based on a bilinear programming approach. The pooling problem occurs frequently in process optimization problems, especially refinery planning models. The main difficulty is that pooling causes an inherent nonlinearity in the otherwise linear models. We shall define the problem by formulating an aggregate mathematical model of a refinery, comment on solution methods for pooling problems that have been presented in the literature, and develop a new method based on convex approximations of the bilinear terms. The method is illustrated on numerical examples
110 citations
TL;DR: In this paper, a necessary and sufficient condition for local optimal solutions of bilevel programming problems is developed using differential stability results for parametric optimization problems, and verification of these conditions reduces to the solution of some auxiliary combinatorial optimization problems.
Abstract: A necessary and a sufficient condition for local optimal solutions of bilevel programming problems are developed using differential stability results for parametric optimization problems. Verification of these conditions reduces to the solution of some auxiliary combinatorial optimization problems.
88 citations
TL;DR: The highest merit factors ever reached for chains of length 81≤N≤201 are found and an evolutionary strategy is introduced, describing the properties of the search algorithm and comparing its results to those of other heuristic methods such as simulated annealing.
Abstract: We investigate skew-symmetric sequences with chain lengths up to N = 71, giving a complete table of all merit factors F≥7 and their associated configurations. We also calculate the exact thermodynami-cal properties of shorter chains (N≤55). We then introduce an evolutionary strategy, describing the properties of our search algorithm and comparing our results to those of other heuristic methods such as simulated annealing. We find the highest merit factors ever reached for chains of length 81≤N≤201.
74 citations
TL;DR: In this paper, a generalized second-order directional derivative and a set-valued generalized Hessian for C 1, 1 functions in real Banach spaces are presented. But the generalized Hessians are not generalized to general functions.
Abstract: In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C1,1 functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gateaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optirnality conditions for mathematical programming problems.
67 citations
TL;DR: In this paper, the operation of the difference of pairs of convex compacta introduced by Demyanov and the related operation proposed by the authors is investigated and the relationship between the Clarke subdifferential and quasidifferential is clarified.
Abstract: Operation of the difference of pairs of convex compacta introduced by Demyanov and the related operation proposed by the authors are investigated. Using these operations the relationship between the Clarke subdifferential and quasidifferential is clarified
64 citations
TL;DR: In this article, a model of competitive financial equilibrium is introduced, which yields the optimal composition of assets and liabilities in each sector's portfolio, as well as the market clearing process.
Abstract: In this paper a model of competitive financial equilibrium is introduced, which yields the optimal composition of assets and liabilities in each sector's portfolio, as well as the market clearing p...
47 citations
TL;DR: In this article, sufficient conditions of minimal character are given in order to obtain the sequential upper and lower semi-continuity of functions defined by: different situations: the unconstrained case, when Y is a general multifunction and when Y(x) is defined by inequalities dependent on x.
Abstract: In convergence spaces (for example topological spaces not necessarily first countable) "sequential" sufficient conditions of minimal character are given in order to obtain the sequential upper and lower semi-continuity of functions defined by: Different situations are considered: the unconstrained case, when Y is a general multifunction and when Y(x) is defined by inequalities dependent on x
38 citations
TL;DR: In this article, the existence of Lagrange multipliers for general Pareto multiobjective mathematical programming problems in Banach spaces is established. But the data are general nonsmooth strongly compactly ipschitzian mappings.
Abstract: We establish the existence of Lagrange multipliers for general Pareto multiobjective mathematical programming problems in Banach spaces. Here the data are general nonsmooth strongly compactly ipschitzian mappings
TL;DR: In this paper, several different definitions of the contingent cone, of the attainable cone and of Clarke's tangent cone are examined in order to state some equivalence relations and the links among them.
Abstract: In the present paper, several different definitions of the contingent cone, of the attainable cone and of Clarke's tangent cone are examined in order to state some equivalence relations and the links among them
TL;DR: In this article, the technique of dimension reduction earlier developed by the first author is applied to the class of nonconvex minimization problems having the so-called rank two property, including the problem of minimizing the product of two affine functions over a polytope.
Abstract: The technique of dimension reduction earlier developed by the first author is applied to the class of nonconvex minimization problems having the so called rank two property. This class includes in particular the problem of minimizing the product of two affine functions over a polytope. An efficient method for solving this class of problems is presented. Also some results of computational experiments with this method are discussed
TL;DR: In this article, the Lagrange duality and scalarization of a cone-subconvex function are studied. And the Lagrangian duality is proven in the context of multiobjective optimization.
Abstract: In this paper, we are concerned with scalarization and the Lagrange duality in multiobjective optimization. After exposing a property of a cone-subconvexlike function, we prove two theorems on scalarization and three theorems of the Lagrange duality.
TL;DR: In this paper, a weak version of Hahn-Banach theorem is derived for convex relations in ordered linear spaces in which a separation theorem of convex sets in [11] is used.
Abstract: In this paper, a weak version of Hahn-Banach theorem is derived for convex relations in ordered linear spaces in which a separation theorem of convex sets in [11] is used. Then this result is applied to derive existence of weak Lagrange multiplier theorem for the following vector optimization problem: where F,H are convex relations from a linear space X to an ordered linear space Y.We also prove existence of weak subgradient for convex relations by using Hahn-Banach theorem. A partial extension of the subgradient formula in [2] to ordered space is obtained
TL;DR: The algorithm generates three sequences which converge, respectively, to optimal solutions for the primal and dual geometric programs, and to a Lagrangian dual solution of the dual geometric program.
Abstract: We present an interior point algorithm for solving the dual geometric programming problem, which avoids nondifferentiability at the boundary, yet uses singular Hessian information. The algorithm generates three sequences which converge, respectively, to optimal solutions for the primal and dual geometric programs, and to a Lagrangian dual solution of the dual geometric program. The sequences are connected by a robust procedure for converting duai GP solutions to primai GP solutions, and error bounds are given. Extensive computational experience is reported including solutions to GP problems having largest known degree of difficulty.
TL;DR: A unified and extended theoretical treatment of the iterative methods for convex minimization design, with emphasis on the mathematical structures relevant for the optimization process, rather than on the statistical background of experimental design.
Abstract: The theory of optimal (approximate) linear regression design has produced several iterative methods to solve a special type of convex minimization problems. The present paper gives a unified and extended theoretical treatment of the methods. The emphasis is on the mathematical structures relevant for the optimization process, rather than on the statistical background of experimental design. So the main body of the paper can be read independently from the experimental design context. Applications are given to a special class of extremum problems arising in statistics. The numerical results obtained indicate that the methods are of practical interest
TL;DR: In this paper, the basic ideas for solving convex non-differentiable minimization problems through piece wise linear approximations to the objective function were discussed, and some variants based on suitable translations of the supporting hyperplanes to the epigraph of the function were introduced.
Abstract: We discuss the basic ideas for solving convex nondifferentiable minimization problems through piece wise linear approximations to the objective function. We revise the traditional cutting plane approach by introducing some variants based on suitable translations of the supporting hyperplanes to the epigraph of the function. We describe also some possible choices for defining a quadratic penalty term on the possible displacement to be added to the model. Finally we report on some numerical experiments on standard test problems
TL;DR: It will be shown that the two methods for solving a linear fractional problem whatever the feasible region is are algorithmically equivalent in the sense that they generate the same finite sequence of points leading to an optimal solution.
Abstract: In this paper two algorithms are suggested for solving a linear fractional problem whatever the feasible region is Such algorithms can be interpreted as a modified version of Martos and Charnes-Cooper algorithms Successively, it will be shown that the two methods are algorithmically equivalent in the sense that they generate the same finite sequence of points leading to an optimal solution This last result can be viewed as an extension of the one given by Wagner-Y4sc:uan[14] for a compact feasible region
TL;DR: The 2-Peripatetic Salesman Problem (2-PSP) minimizes the total length of 2 edge-disjoint Hamil-tonian cycles, which arises in designing communication or computer networks, or whenever one aims to increase network reliability using disjoint tours.
Abstract: The 2-Peripatetic Salesman Problem (2-PSP) minimizes the total length of 2 edge-disjoint Hamil-tonian cycles. This type of problems arises in designing communication or computer networks, or whenever one aims to increase network reliability using disjoint tours. The NP-hardness of the 2-PSP is shown. Lower bound values are obtained by generalizing the 1-tree approach for the TSP to a 2 edge-disjoint 1-trees approach for the 2-PSP. One can construct 2 edge-disjoint 1-trees using a greedy algorithm, into which a partitioning procedure is incorporated that runs O(n 2 log n) time. Upper bound solutions are obtained by two heuristics based on a lower bound solution and by a modified Savings heuristic for problems up to 140 cities.
TL;DR: In this article, Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program.
Abstract: Three generalizations of the criss-cross method for quadratic programming are presented here. Tucker’s, Cottle’s and Dantzig’s principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program A finite criss-cross method, based on least-index resolution, is constructed for solving the LCP. In proving finiteness, orthogonality properties of pivot tableaus and positive semidefiniteness of quadratic matrices are used In the last section some special cases and two further variants of the quadratic criss-cross method are discussed. If the matrix of the LCP has full rank, then a surprisingly simple algorithm follows, which coincides with Murty’s ‘Bard type schema’ in the P matrix case
TL;DR: In this paper, the authors give explicit sequential conditions on convergence of sequences (fn ) and (Yn ) in order to obtain sequential convergence properties of the sequence of marginal functions, where (n)n is a sequence of extended real valued functions defined on the product U×V of two convergence spaces and (yn )n is defined on U and valued in non empty parts of V.
Abstract: Let (fn )n be a sequence of extended real valued functions defined on the product U×V of two convergence spaces and let (Yn ) be a sequence of multifunctions defined on U and valued in non empty parts of V. Aim of this paper is to give explicit sequential conditions on convergence of sequences (fn ) and (Yn ) in order to obtain sequential convergence properties of the sequence of marginal functions:
TL;DR: A new class of continuously differentiable globally exact penalty functions for the solution of minimization problems with simple bounds on some (all) of the variables is proposed.
Abstract: In this paper we propose a new class of continuously differentiable globally exact penalty functions for the solution of minimization problems with simple bounds on some (all) of the variables. The penalty functions in this class fully exploit the structure of the problem and are easily computable. Furthermore we introduce a simple updating rule for the penalty parameter that can be used in conjunction with unconstrained minimization techniques to solve the original problem.
TL;DR: A special class of relations is common in many of optimization problems as discussed by the authors, and they extract it in a general form and provide sufficient conditions for the existence of maximal points in the general form.
Abstract: A special class of relations is common in many of optimization problems. We extract it in a general form and provide sufficient conditions for the existence of maximal points. The general results a...
TL;DR: In this article, an exact penalty function for optimization problems defined by subanalytic functions with equality or inequality constraints is defined, where the penalty function is defined by a sub-analytic function with equality constraints.
Abstract: We define an exact penalty function for optimization problems defined by subanalytic functions with equality or inequality constraints
TL;DR: The proposed technique proceeds from a user defined number of linearized constraints, that is to be used internally to determine the size of the quadratic programming subproblem, and significant constraints are then selected automatically by the algorithm.
Abstract: For solving the smooth constrained nonlinear programming problem, sequential quadratic programming (SQP) methods are considered to be the standard tool, as long as they are applicable. However one possible situation preventing the successful solution by a standard SQP-technique, arises if problems with a very large number of constraints are to be solved. Typical applications are semi-infinite or min-max optimization, optimal control or mechanical structural optimization. The proposed technique proceeds from a user defined number of linearized constraints, that is to be used internally to determine the size of the quadratic programming subproblem. Significant constraints are then selected automatically by the algorithm. Details of the numerical implementation and some experimental results are presented
TL;DR: In this article, the authors deal with generalizations of the usual convexity of real-valued functions in such a manner that "convex" is extended to " -Convex", and -conveXity is required only on straight lines with directions from a given cone K. Under certain assumptions on the generating family and on K, local boundedness and continuity properties are obtained.
Abstract: This article deals with generalizations of the usual convexity of real-valued functions in such a manner that “convex” is extended to “ -convex” and -convexity is required only on straight lines with directions from a given cone K. Under certain assumptions on the generating family and on K, for functions of such kind (called -convex on K-lines) local boundedness and continuity properties are obtained. The main results are applied to a number of examples. In particular, Morrey’s rank 1 convexity and a special type of “rough convexity” are considered
TL;DR: In this paper, a method for solving these problems is suggested, a motivating example from service-point-location problems is described, and a method to solve the problem is presented.
Abstract: Optimization problems of the form subject to are considered. In these problems are continuous and are continuous and upper unimodal on [hj, Hj ] for all i, j A method for solving these problems is suggested, a motivating example from service-point-location problems is described
TL;DR: In this article, the generalization of the inexact linear programming is introduced and its properties are investigated, and the special case of the linear program is the well known linear programming.
Abstract: In this paper the generalization of the inexact programming is introduced and next its properties are investigated. The special case of the inexact linear programming is the well known linear programming
TL;DR: All the members in the family of algorithms with {⊘ k } chosen so that for some and all kare locally and q-superlinearly convergent are extensions of the Broyden family of approximants of the objective function Hessian.
Abstract: Ariyawansa [3] has presented a class of coilinear scaling algorithms for unconstrained minimization. A certain family of algorithms contained in this class may be considered as an extension of the family of quasi-Newton methods with the Broyden family of approximants of the objective function Hessian. Let ⊘ k be the parameter that specifies the Broyden family member used at the (k+1)th iterate, .In [3], the members in this family of collinear scaling algorithms corresponding to ⊘ k := 0 for all k(which extends the DFP method) and ⊘ k :=1 for all k(which extends the BFGS method) were shown to be locally and q-superlinearly convergent. In this paper, we extend that result to other members in the above family of coilinear scaling algorithms in the following sense:all the members in the family of algorithms with {⊘ k } chosen so that for some and all kare locally and q-superlinearly convergent