Showing papers in "Journal of Optimization Theory and Applications in 1992"
TL;DR: In this article, it was shown that the coordinate descent method is convergent for the symmetric monotone linear complementarity problem, where the cost function is the composition of an affine mapping with a strictly convex function which is twice differentiable in its effective domain.
Abstract: The coordinate descent method enjoys a long history in convex differentiable minimization. Surprisingly, very little is known about the convergence of the iterates generated by this method. Convergence typically requires restrictive assumptions such as that the cost function has bounded level sets and is in some sense strictly convex. In a recent work, Luo and Tseng showed that the iterates are convergent for the symmetric monotone linear complementarity problem, for which the cost function is convex quadratic, but not necessarily strictly convex, and does not necessarily have bounded level sets. In this paper, we extend these results to problems for which the cost function is the composition of an affine mapping with a strictly convex function which is twice differentiable in its effective domain. In addition, we show that the convergence is at least linear. As a consequence of this result, we obtain, for the first time, that the dual iterates generated by a number of existing methods for matrix balancing and entropy optimization are linearly convergent.
589 citations
TL;DR: In this article, a vector variational inequality is studied under convexity assumptions and without convexness assumptions, and existence theorems for solutions under these assumptions are given.
Abstract: A vector variational inequality is studied. The paper deals with existence theorems for solutions under convexity assumptions and without convexity assumptions.
266 citations
TL;DR: This paper replaces quadratic additive terms in the objectives of the subproblems with more general D-functions which resemble (but are not strictly) distance functions which, when used in the proximal minimization algorithm, preserve its overall convergence.
Abstract: The original proximal minimization algorithm employs quadratic additive terms in the objectives of the subproblems. In this paper, we replace these quadratic additive terms by more generalD-functions which resemble (but are not strictly) distance functions. We characterize the properties of suchD-functions which, when used in the proximal minimization algorithm, preserve its overall convergence. The quadratic case as well as an entropy-oriented proximal minimization algorithm are obtained as special cases.
264 citations
TL;DR: In this paper, it is shown that under appropriate technical assumptions, a Pareto-efficient allocationX maximizes the total benefit relative to the utility levels it yields, whereas if an allocationX yields zero benefit and maximizes a total benefit function, then that allocation is Pare-to efficient, and this in turn is equivalent, under appropriate assumptions, to a competitive equilibrium.
Abstract: This paper develops several optimization principles relating the fundamental concepts of Pareto efficiency and competitive equilibria. The beginning point for this development is the introduction of a new function describing individual preferences, closely related to willingness-to-pay, termed the benefit function. An important property of the benefit function is that it can be summed across individuals to obtain a meaningful measure of total benefit relative to a given set of utility levels; and the optimization principles presented in the paper are based on maximization of this total benefit.
Specifically, it is shown that, under appropriate technical assumptions, a Pareto-efficient allocationX maximizes the total benefit relative to the utility levels it yields. Conversely, if an allocationX yields zero benefit and maximizes the total benefit function, then that allocation is Pareto efficient. The Lagrange multipliersp of the benefit maximization problem serve as prices; and the (X,p) pair satisfies a generalized saddle-point property termed a Lagrange equilibrium. This in turn is equivalent, under appropriate assumptions, to a competitive equilibrium.
There are natural duals to all of the results stated above. The dual optimization principle is based on a surplus function which is a function of prices. The surplus is the total income generated at pricesp, minus the total income required to obtain given utility levels. The dual optimization principle states that prices that are dual (or indirect) Pareto efficient minimize total surplus and render it zero. Conversely, a set of prices that minimizes total surplus and renders it zero is a dual Pareto efficient set of prices.
The results of the paper can be viewed as augmenting the first and second theorems of welfare economics (and their duals) to provide a family of results that relate the important economic concepts of Pareto efficiency, equilibrium, dual (or indirect) Pareto efficiency, total benefit, Lagrange equilibrium, and total surplus.
263 citations
TL;DR: In this paper, some known existence results for a nonlinear complementarity problem in a finite-dimensional Hilbert space are generalized to an infinite-dimensional real Hilbert space and applications to a class of nonlinear complearity problems and the study of the post-critical equilibrium state of a thin elastic plate subjected to unilateral conditions are given.
Abstract: In this paper, some existence results for a nonlinear complementarity problem involving a pseudo-monotone mapping over an arbitrary closed convex cone in a real Hilbert space are established. In particular, some known existence results for a nonlinear complementarity problem in a finite-dimensional Hilbert space are generalized to an infinite-dimensional real Hilbert space. Applications to a class of nonlinear complementarity problems and the study of the post-critical equilibrium state of a thin elastic plate subjected to unilateral conditions are given.
215 citations
TL;DR: An extension of the well-known Mangasarian-Fromovitz constraint qualification (EMFCQ) is introduced and the main result is the equivalence of the topological stability of the feasible setM[H, G] and the validity of EMFCQ.
Abstract: The problem of the minimization of a functionf: ℝn→ℝ under finitely many equality constraints and perhaps infinitely many inequality constraints gives rise to a structural analysis of the feasible setM[H, G]={x∈ℝn¦H(x)=0,G(x, y)≥0,y∈Y} with compactY⊂ℝr. An extension of the well-known Mangasarian-Fromovitz constraint qualification (EMFCQ) is introduced. The main result for compactM[H, G] is the equivalence of the topological stability of the feasible setM[H, G] and the validity of EMFCQ. As a byproduct, we obtain under EMFCQ that the feasible set admits local linearizations and also thatM[H, G] depends continuously on the pair (H, G). Moreover, EMFCQ is shown to be satisfied generically.
96 citations
TL;DR: This paper presents a nonadjacent extreme-point search algorithm for finding a globally optimal solution for problem (P), and finds an exact extreme- point optimal solution after a finite number of iterations.
Abstract: The problem (P) of optimizing a linear function over the efficient set of a multiple-objective linear program serves many useful purposes in multiple-criteria decision making. Mathematically, problem (P) can be classified as a global optimization problem. Such problems are much more difficult to solve than convex programming problems. In this paper, a nonadjacent extreme-point search algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact extreme-point optimal solution for the problem after a finite number of iterations. It can be implemented using only linear programming methods. Convergence of the algorithm is proven, and a discussion is included of its main advantages and disadvantages.
79 citations
TL;DR: In this paper, the authors proved that the set of feasible points for which MFCQ essentially differs from LICQ is small in a specified sense, and showed that the constraint set (even in semi-infinite optimization) is locally representable in epigraph form.
Abstract: The linear independence constraint qualification (LICQ) and the weaker Mangasarian-Fromovitz constraint qualification (MFCQ) are well-known concepts in nonlinear optimization. A theorem is proved suggesting that the set of feasible points for which MFCQ essentially differs from LICQ is small in a specified sense. As an auxiliary result, it is shown that, under MFCQ, the constraint set (even in semi-infinite optimization) is locally representable in epigraph form.
72 citations
TL;DR: In this paper, the orthogonal projection of the current point onto a hyperplane corresponds to a surrogate constraint which is constructed through a positive combination of a group of violated constraints.
Abstract: New iterative methods for solving systems of linear inequalities are presented Each step in these methods consists of finding the orthogonal projection of the current point onto a hyperplane corresponding to a surrogate constraint which is constructed through a positive combination of a group of violated constraints Both sequential and parallel implementations are discussed
65 citations
TL;DR: In this paper, the authors studied dual functionals which have good asymptotical behavior and minimal conditions for the convergence of the Gauss-Seidel methods are given and applied to such kinds of functionals.
Abstract: We study dual functionals which have two fundamental properties. Firstly, they have a good asymptotical behavior. Secondly, to each dual sequence of subgradients converging to zero, one can associate a primal sequence which converges to an optimal solution of the primal problem. Furthermore, minimal conditions for the convergence of the Gauss-Seidel methods are given and applied to such kinds of functionals.
65 citations
TL;DR: It is proved that the number of iterations required by the algorithm to converge to an ε-optimal solution isO((1+M2)n∣logε∣), depending on the updating scheme for the lower bound.
Abstract: In this paper, we describe a natural implementation of the classical logarithmic barrier function method for smooth convex programming. It is assumed that the objective and constraint functions fulfill the so-called relative Lipschitz condition, with Lipschitz constantM>0.
TL;DR: In this article, the authors describe a natural implementation of the classical logarithmic barrier function method for smooth convex programming, where the objective and constraint functions are assumed to be the same.
Abstract: In this paper, we describe a natural implementation of the classical logarithmic barrier function method for smooth convex programming. It is assumed that the objective and constraint functions ful...
TL;DR: In this article, the authors reconstruct a proof of a classical result due to Hardy and Littlewood, which is not covered by the Hardy-Littlewood theorem, and provide either examples or complete citations for other related cases which are not covered.
Abstract: In this note, we reconstruct a proof of a classical result due to Hardy and Littlewood. While this result has played an important role in the modern theories of Markov decision processes and stochastic games, it is not that easy to find its proof in the literature in the format in which it has been applied. Furthermore, we supply either examples or complete citations for the other related cases which are not covered by the Hardy-Littlewood theorem.
TL;DR: In this article, a non-calarized vector cost function is used to solve vector optimization problems with non-conconcilable objectives, which is made possible due to the ability to obtain full global optimal solutions.
Abstract: A new approach to multiobjective optimization is presented which is made possible due to our ability to obtain full global optimal solutions. A distinctive feature of this approach is that a vector cost function is nonscalarized. The method provides a means for the solution of vector optimization problems with nonreconcilable objectives.
TL;DR: In this article, an active set strategy with three types of constraints (nonactive, semi-active, and active) is proposed to solve the min-max problem, where nonactive constraints are treated as equality constraints and semiactive constraints as inequality constraints and assigned slack variables.
Abstract: The purpose of this paper is to suggest a new, efficient algorithm for the minmax problem $$\mathop {min}\limits_x \mathop {max}\limits_i f_i (x), x \in \Re ^n , i = 1, \ldots ,m,$$ wheref i ,i=1,...,m, are real-valued functions defined on ? n . The problem is transformed into an equivalent inequality-constrained minimization problem, mint, s.t.f i (x)?t≤0, for alli, i=1,...,m. The algorithm has these features: an active-set strategy with three types of constraints; the use of slack variables to handle inequality constraints; and a trust-region strategy taking advantage of the structure of the problem. Following Tapia, this problem is solved by an active set strategy which uses three types of constraints (called here nonactive, semiactive, and active). Active constraints are treated as equality constraints, while semiactive constraints are treated as inequality constraints and are assigned slack variables. This strategy helps to prevent zigzagging. Numerical results are provided.
TL;DR: In this article, a general auxiliary principle technique is used to suggest and analyze a novel and innovative iterative algorithm for solving variational inequalities and optimization problems, and the convergence criteria are discussed.
Abstract: In this paper, we consider a general auxiliary principle technique to suggest and analyze a novel and innovative iterative algorithm for solving variational inequalities and optimization problems. We also discuss the convergence criteria.
TL;DR: In this article, proper solutions in the Kuhn-Tucker sense for multiobjective mathematical programming problems with parameters in infinite-dimensional spaces are defined and compared with other definitions via suitable representatives: the Benson, Geoffrion, and Hurwicz properness.
Abstract: We define proper solutions in the Kuhn-Tucker sense for multiobjective mathematical programming problems with parameters in infinite-dimensional spaces and compare them with other definitions via suitable representatives: the Benson, Geoffrion, and Hurwicz properness. Necessary and/or sufficient conditions for proper solutions are proved. Problems with and without constraint qualifications are considered under relaxed convexity and differentiability assumptions.
TL;DR: Convexlike and concavelike conditions are of interest for extensions of the Von Neumann minimax theorem as discussed by the authors, and these conditions also play a certain role in deriving generalized alternative theorems of the Gordan, Motzkin, and Farkas type and Lagrange multiplier results for constrained minimization problems.
Abstract: Convexlike and concavelike conditions are of interest for extensions of the Von Neumann minimax theorem. Since the beginning of the 80's, these conditions also play a certain role in deriving generalized alternative theorems of the Gordan, Motzkin, and Farkas type and Lagrange multiplier results for constrained minimization problems.
TL;DR: A method for stiffness matrix adjustment which preserves the sparsity pattern of an original matrix, requires comparatively modest computational resources, and allows robust handling of noisy modal data is introduced.
Abstract: Problems of model correlation and system identification are central in the design, analysis, and control of large space structures. Of the numerous methods that have been proposed, many are based on finding minimal adjustments to a model matrix sufficient to introduce some desirable quality into that matrix. In this work, several of these methods are reviewed, placed in a modern framework, and linked to other previously known ideas in computational linear algebra and optimization. This new framework provides a point of departure for a number of new methods which are introduced here. Significant among these is a method for stiffness matrix adjustment which preserves the sparsity pattern of an original matrix, requires comparatively modest computational resources, and allows robust handling of noisy modal data. Numerical examples are included to illustrate the methods presented herein.
TL;DR: This paper considers a class of nonlinear minimum-maximum optimization problems subject to boundedness constraints on the decision vectors and develops three algorithms for finding the min-max point using the concept of solving an associated dynamical system.
Abstract: In this paper, we consider a class of nonlinear minimum-maximum optimization problems subject to boundedness constraints on the decision vectors. Three algorithms are developed for finding the min-max point using the concept of solving an associated dynamical system. In the first and third algorithms, solutions are obtained by solving systems of differential equations. The second algorithm is a discrete version of the first algorithm. The trajectories generated by the first and second algorithms may move inside or on the boundary of the constraint set, while the third algorithm ensures that any trajectory that begins inside the constraint region remains in its interior. Sufficient conditions for global convergence of the two algorithms are also established. For illustration, four numerical examples are solved.
TL;DR: In this paper, a reference point approximation algorithm is presented for the interactive solution of bicriterial nonlinear optimization problems with inequality and equality constraints, where the decision maker may choose arbitrary reference points in the criteria space.
Abstract: This paper presents a reference point approximation algorithm which can be used for the interactive solution of bicriterial nonlinear optimization problems with inequality and equality constraints. The advantage of this method is that the decision maker may choose arbitrary reference points in the criteria space. Moreover, a special tunneling technique is given for the computation of global solutions of certain subproblems. Finally, the proposed method is applied to a mathematical example and a problem in mechanical engineering.
TL;DR: Recently, Zhang, Tapia, and Dennis as mentioned in this paper presented a superlinear and quadratic convergence theory for the duality gap sequence in primal-dual interior-point methods for linear programming.
Abstract: Recently, Zhang, Tapia, and Dennis (Ref. 1) produced a superlinear and quadratic convergence theory for the duality gap sequence in primal-dual interior-point methods for linear programming. In this theory, a basic assumption for superlinear convergence is the convergence of the iteration sequence; and a basic assumption for quadratic convergence is nondegeneracy. Several recent research projects have either used or built on this theory under one or both of the above-mentioned assumptions. In this paper, we remove both assumptions from the Zhang-Tapia-Dennis theory.
TL;DR: In this paper, the minimum-time problem to reach aC 1-manifold target was considered and it was shown that the minimum time function is locally Lipschitz and respectively 1/2-Holder continuous.
Abstract: We consider the minimum-time problem to reach aC 1-manifold target. Using implicit function techniques, under conditions of order 0 and 1 on the vector fields, we prove that the minimum-time function is locally Lipschitz and respectively 1/2-Holder continuous.
TL;DR: A new generalized Polak-Ribière conjugate gradient algorithm is proposed for unconstrained optimization, and its numerical and theoretical properties are discussed.
Abstract: A new generalized Polak-Ribiere conjugate gradient algorithm is proposed for unconstrained optimization, and its numerical and theoretical properties are discussed. The new method is, in fact, a particular type of two-dimensional Newton method and is based on a finite-difference approximation to the product of a Hessian and a vector.
TL;DR: A global optimization approach is developed where random starting conditions are improved by using special line searches to determine the optimum in nonlinear optimal control problems.
Abstract: To determine the optimum in nonlinear optimal control problems, it is proposed to convert the continuous problems into a form suitable for nonlinear programming (NLP). Since the resulting finite-dimensional NLP problems can present multiple local optima, a global optimization approach is developed where random starting conditions are improved by using special line searches. The efficiency, speed, and reliability of the proposed approach is examined by using two examples.
TL;DR: In this article, a class of univariate functions is exhibited for which the global optimum will be missed when using such a procedure, even if the multiple of the largest slope between successive evaluation points is arbitrarily large.
Abstract: Several authors have proposed estimating Lipschitz constants in global optimization by a multiple of the largest slope (in absolute value) between successive evaluation points. A class of univariate functions is exhibited for which the global optimum will be missed when using such a procedure, even if the multiple is arbitrarily large.
TL;DR: In this paper, an optimal control problem for then-dimensional diffusion equation with a sequence of Radon measures as generalized control variables was studied, where a desired final state is not reachable.
Abstract: The present paper is concerned with an optimal control problem for then-dimensional diffusion equation with a sequence of Radon measures as generalized control variables. Suppose that a desired final state is not reachable. We enlarge the set of admissible controls and provide a solution to the corresponding moment problem for the diffusion equation, so that the previously chosen desired final state is actually reachable by the action of a generalized control. Then, we minimize an objective function in this extended space, which can be characterized as consisting of infinite sequences of Radon measures which satisfy some constraints. Then, we approximate the action of the optimal sequence by that of a control, and finally develop numerical methods to estimate these nearly optimal controls. Several numerical examples are presented to illustrate these ideas.
TL;DR: In this paper, the concept of state-dependent weights is studied, which offers the flexibility to approximate the changing weights of the decision maker, and the relationship between statedependent weights and value functions is discussed.
Abstract: In this paper, the concept of state-dependent weights is studied. This offers the flexibility to approximate the changing weights of the decision maker. The relationship between state-dependent weights and value functions is discussed.
TL;DR: A criterion recently developed by Hiriart-Urruty and Lemarechal is thoroughly examined in the case of concave quadratic problems and reformulated into copositivity conditions for global optimality.
Abstract: Second-order necessary and sufficient conditions for local optimality in constrained optimization problems are discussed. For global optimality, a criterion recently developed by Hiriart-Urruty and Lemarechal is thoroughly examined in the case of concave quadratic problems and reformulated into copositivity conditions.
TL;DR: In this paper, a control problem where the coefficients of the linear dynamics are functions of a noisily observed Markov chain is considered, and a minimum principle and a new equation for an adjoint process are obtained.
Abstract: A control problem is considered where the coefficients of the linear dynamics are functions of a noisily observed Markov chain. The approximation introduced is to consider these coefficients as functions of the filtered estimate of the state of the chain; this gives rise to a finite-dimensional conditional Kalman filter. A minimum principle and a new equation for an adjoint process are obtained.