Showing papers on "Convex optimization published in 1976"
Book•
01 Jan 1976TL;DR: In this article, the authors consider non-convex variational problems with a priori estimate in convex programming and show that they can be solved by the minimax theorem.
Abstract: Preface to the classics edition Preface Part I. Fundamentals of Convex Analysis. I. Convex functions 2. Minimization of convex functions and variational inequalities 3. Duality in convex optimization Part II. Duality and Convex Variational Problems. 4. Applications of duality to the calculus of variations (I) 5. Applications of duality to the calculus of variations (II) 6. Duality by the minimax theorem 7. Other applications of duality Part III. Relaxation and Non-Convex Variational Problems. 8. Existence of solutions for variational problems 9. Relaxation of non-convex variational problems (I) 10. Relaxation of non-convex variational problems (II) Appendix I. An a priori estimate in non-convex programming Appendix II. Non-convex optimization problems depending on a parameter Comments Bibliography Index.
4,434 citations
TL;DR: The theory of the proximal point algorithm for maximal monotone operators is applied to three algorithms for solving convex programs, one of which has not previously been formulated and is shown to have much the same convergence properties, but with some potential advantages.
Abstract: The theory of the proximal point algorithm for maximal monotone operators is applied to three algorithms for solving convex programs, one of which has not previously been formulated Rate-of-convergence results for the “method of multipliers,” of the strong sort already known, are derived in a generalized form relevant also to problems beyond the compass of the standard second-order conditions for oplimality The new algorithm, the “proximal method of multipliers,” is shown to have much the same convergence properties, but with some potential advantages
1,221 citations
TL;DR: Multivalued functions with convex graphs are shown to exhibit certain desirable regularity properties when their ranges have internal points to develop a perturbation theory for convex inequalities and to extend results on the continuity of convex functions.
Abstract: Multivalued functions with convex graphs are shown to exhibit certain desirable regularity properties when their ranges have internal points. These properties are applied to develop a perturbation theory for convex inequalities and to extend results on the continuity of convex functions.
348 citations
TL;DR: A new method designed to globally minimize concave functions over linear polyhedra is described, and an example problem is solved, and computational considerations are discussed.
Abstract: A new method designed to globally minimize concave functions over linear polyhedra is described. Properties of the method are discussed, an example problem is solved, and computational considerations are discussed.
169 citations
TL;DR: In this paper, a duality theory for linear and concave-convex fractional programs is developed and related to recent results by Bector, Craven-Mond, Jagannathan, Sharma-Swarup, et al.
Abstract: This paper, which is presented in two parts, is a contribution to the theory of fractional programming, i.e., maximization of quotients subject to constraints. In Part I a duality theory for linear and concave-convex fractional programs is developed and related to recent results by Bector, Craven-Mond, Jagannathan, Sharma-Swarup, et al. Basic duality theorems of linear, quadratic and convex programming are extended. In Part II Dinkelbach's algorithm solving fractional programs is considered. The rate of convergence as well as a priori and a posteriori error estimates are determined. In view of these results the stopping rule of the algorithm is changed. Also the starting rule is modified using duality as introduced in Part I. Furthermore a second algorithm is proposed. In contrast to Dinkelbach's procedure the rate of convergence is still controllable. Error estimates are obtained too.
150 citations
Book•
01 Jan 1976
TL;DR: In this paper, the authors present a generalization of Convex Analysis to nonlinear programming via conjugate functions, including linear subspaces and affine manifolds.
Abstract: I: Convex Analysis.- 1: Linear Subspaces and Affine Manifolds.- 1.1 Linear Subspaces and Orthogonal Complements.- 1.2 Linear Independence and Dimensionality.- 1.3 Projection Theorem.- 1.4 Affine Manifolds.- 2: Convex Sets.- 2.1 Convex Cones, Convex Sets and Convex Hills.- 2.2 Caratheodory Type Theorems.- 2.3 Relative Interior and Related Properties of Convex Sets.- 2.4 Support and Separation Theorems.- 3: Convex Cones.- 3.1 Cones, Convex Cones and Polar Cones.- 3.2 Polyhedral Cones.- 3.3 Cones Generated by Sets.- 3.4 Cone of Tangents.- 3.5 Cone of Attainable Directions, Cone of Feasible Directions and Cone of Interior Directions.- 4: Convex Functions.- 4.1 Definitions and Preliminary Results.- 4.2 Continuity and Directional Differentiability of Convex Functions.- 4.3 Differentiable Convex Functions.- 4.4 Some Examples of Convex Functions.- 4.5 Generalization of Convex Functions.- II: Optimality Conditions and Duality.- 5: Stationary Point Optimality Conditions with Differentiability.- 5.1 Inequality Constrained Problems.- 5.2 Inequality and Equality Constrained Problems.- 5.3 Optimality Criteria of the Minimum Principle Type.- 6: Constraint Qualifications.- 6.1 Inequality Constrained Problems.- 6.2 Equality and Inequality Constrained Problems.- 6.3 Necessary and Sufficient Qualification.- 7: Convex Programming without Differentiability.- 7.1 Saddle Point Optimality Criteria.- 7.2 Stationary Point Optimality Conditions.- 8: Lagrangian Duality.- 8.1 Definitions and Preliminary Results.- 8.2 The Strong Duality Theorem.- 9: Conjugate Duality.- 9.1 Closure of a Function.- 9.2 Conjugate Functions.- 9.3 Main Duality Theorem.- 9.4 Nonlinear Programming via Conjugate Functions.- Selected References.
128 citations
TL;DR: It is established that a large class of problems of finding optimal facility locations on an imbedding of a finite, undirected network having positive arc lengths is convex, in a well defined sense, for all choices of the data if and only if the network is a tree.
Abstract: This paper studies problems of finding optimal facility locations on an imbedding of a finite, undirected network having positive arc lengths. We establish that a large class of such problems is convex, in a well defined sense, for all choices of the data if and only if the network is a tree. A number of useful properties of related convex functions end convex sets are identified.
96 citations
TL;DR: In this article, the basic dual problem and extended dual problem associated with a two-stage stochastic program are shown to be equivalent, if the program is strictly feasible and satisfies a condition generalizing, in a sense, the condition of relatively complete recourse.
Abstract: The basic dual problem and extended dual problem associated with a two-stage stochastic program are shown to be equivalent, if the program is strictly feasible and satisfies a condition generalizing, in a sense, the condition of relatively complete recourse in stochastic linear programming. Combined with earlier results, this yields the fact that, under the same assumptions, solutions to the program can be characterized in terms of saddle points of the basic Lagrangian. A couple of examples are used to illustrate the salient points of the theory. The last section contains a review of the principal implications of the results of this paper combined with those of three preceding papers also devoted to stochastic convex programs.
95 citations
93 citations
TL;DR: A class of combined primal–dual and penalty methods for constrained minimization which generalizes the method of multipliers is proposed and analyzed and it is shown that the rate of convergence may be linear or superlinear with arbitrary Q-order of convergence depending on the problem at hand.
Abstract: In this paper we propose and analyze a class of combined primal–dual and penalty methods for constrained minimization which generalizes the method of multipliers. We provide a convergence and rate of convergence analysis for these methods for the case of a convex programming problem. We prove global convergence in the presence of both exact and inexact unconstrained minimization, and we show that the rate of convergence may be linear or superlinear with arbitrary Q-order of convergence depending on the problem at hand and the form of the penalty function employed.
79 citations
TL;DR: In this article, a two-stage stochastic programming problem with recourse is studied in terms of an extended Lagrangian function which allows certain multipliers to be elements of a dual space (i?00)*, rather than an ϊ£λ space.
Abstract: A two-stage stochastic programming problem with recourse is studied here in terms of an extended Lagrangian function which allows certain multipliers to be elements of a dual space (i?00)*, rather than an ϊ£λ space. Such multipliers can be decomposed into an i^-component and a "singular" component. The generalization makes it possible to characterize solutions to the problem in terms of a saddle-point, if the problem is strictly feasible. The Kuhn-Tucker conditions for the basic duality framework are modified to admit singular multipliers. It is shown that the optimal multiplier vectors in the extended dual problem are, in at least one broad case, ideal limits of maximizing sequences of multiplier vectors in the basic dual problem.
TL;DR: The optimality conditions are stated in terms of consistency or inconsistency of a family of systems of linear inequalities and cone relations.
Abstract: Necessary and sufficient conditions of optimality are given for convex programming problems with no constraint qualification. The optimality conditions are stated in terms of consistency or inconsistency of a family of systems of linear inequalities and cone relations.
TL;DR: In this paper, the authors discuss convex quadratic programming problems with variable coefficients in the linear part of the objective function or/and in the right hand side of the constraints.
Abstract: In this paper we discuss convex quadratic programming problems with variable coefficients in the linear part of the objective function or/and in the right hand side of the constraints. Local and global stability statements are contained. An important global stability theorem is proved for a feneral non-linear programming problem arbitrary, where F is a continuous function over is a nonempty compact subset of E n . A possibility of calculating of a local stability set for the convex quadratic parametric programming problem is also given. This method is not based on an algorithm for quadratic programming problems.
TL;DR: In this article, the optimal impulsive control of systems arising from linear compartment models for drug distribution in the human body is considered and a constructive algorithm for the computation of the optimal control relative to a convex cost functional is developed.
Abstract: Optimal impulsive control of systems arising from linear compartment models for drug distribution in the human body is considered. A system of linear, time-invariant, homogeneous differential equations is given along with a set of continuous constraints on state and control. The object is to develop a constructive algorithm for the computation of the optimal control relative to a convex cost functional. It is first shown that under suitable hypotheses, satisfying the continuous constraints is equivalent to satisfying the constraints at a finite set of abstractly definedcritical points. Once these critical points have been determined, the solution of the optimal control problem is found as the solution of a finite-dimensional convex programming problem. The set of critical points can often be determineda priori solely from the qualitative behavior of the solutions of the system. A class of such problems, generalizing the so-calledplateau effect, is considered in detail. It is shown that the solution achieving the plateau effect is indeed optimal in certain cases. In a subsequent paper, an iterative algorithm will be given for the solution of these problems when the critical points cannot all be determineda priori.
Journal Article•
TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.org/conditions) of the agreement with the Scuola Normale Superiore di Pisa are defined.
Abstract: © Scuola Normale Superiore, Pisa, 1976, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
TL;DR: In this paper, the Fritz John necessary conditions for optimality of a differentiable nonlinear programming problem have been shown, given additional convexity hypotheses, to be also sufficient (by Gulati, Craven, and others).
Abstract: The Fritz John necessary conditions for optimality of a differentiable nonlinear programming problem have been shown, given additional convexity hypotheses, to be also sufficient (by Gulati, Craven, and others). This sufficiency theorem is now extended to minimization (suitably defined) of a function taking values in a partially ordered space, and to (convex) objective and constraint functions which are not always differentiable. The results are expressed in terms of subgradients.
TL;DR: The problem of determining the flows in the pipes of a general hydraulic network is shown to be equivalent to either of a pair of convex programming problems with linear constraints via the theory of (generalized) Geometric Programming.
Abstract: The problem of determining the flows (and pressures) in the pipes of a general hydraulic network, for given input and output flows and/or given input and output pressure heads, is shown to be equivalent to either of a pair of convex programming problems with linear constraints. This is accomplished via the theory of (generalized) Geometric Programming. The equivalence of these problems is exploited to prove existence and uniqueness of the flow solution under certain conditions as well as to derive an algorithm which calculates this solution. Computational aspects of implementing the algorithm are considered in some detail and results obtained for a general example problem are presented. A brief discussion of the application of the methods of the paper to problems in electrical network analysis, transportation network analysis and the elastic analysis of structural trusses, is also given.
TL;DR: Conditions are given which are sufficient to prove the existence of a unique unconstrained minimizer in a convex compact set.
Abstract: Conditions are given which are sufficient to prove the existence of a unique unconstrained minimizer in a convex compact set. An exact formula is given for the amount by which a value of the function exceeds its global minimum.
TL;DR: The multistage or staircase structure appears naturally in many models with time horizons and when the nature of the problem presents certain structural forms, the decomposition allows for the introduction of more efficient techniques.
Abstract: The multistage or staircase structure appears naturally in many models with time horizons. This paper presents and discusses a method for decomposition when the problem functions are convex. Among the techniques which can be used to solve the subproblems are the Dantzig–Wolfe convex programming algorithm and Bender’s decomposition. Furthermore, when the nature of the problem presents certain structural forms, the decomposition allows for the introduction of more efficient techniques.
TL;DR: The method of centers is a well-known method for solving nonlinear programming problems having inequality constraints as discussed by the authors, and it is known to converge linearly for finite-dimensional convex programs for which the objective function has a positive-definite Hessian near the solution.
Abstract: The method of centers is a well-known method for solving nonlinear programming problems having inequality constraints. Pironneau and Polak have recently presented a new version of this method. In the new method, the direction of search is obtained, at each iteration, by solving a convex quadratic programming problem. This direction finding subprocedure is essentially insensitive to the dimension of the space on which the problem is defined. Moreover, the method of Pironneau and Polak is known to converge linearly for finite-dimensional convex programs for which the objective function has a positive-definite Hessian near the solution (and for which the functions involved are twice continuously differentiable). In the present paper, the method and a completely implementable version of it are shown to converge linearly for a very general class of finite-dimensional problems; the class is determined by a second-order sufficiency condition and includes both convex and nonconvex problems. The arguments employed here are based on the indirect sufficiency method of Hestenes. Furthermore, the arguments can be modified to prove linear convergence for a certain class of infinite-dimensional convex problems, thus providing an answer to a conjecture made by Pironneau and Polak.
TL;DR: Necessary and sufficient conditions for optimality are given, for convex programming problems, without constraint qualification, in terms of a single mathematical program, which can be chosen to be bilinear.
Abstract: Necessary and sufficient conditions for optimality are given, for convex programming problems, without constraint qualification, in terms of a single mathematical program, which can be chosen to be bilinear.
TL;DR: In this article, the convergence of a certain class of monotonic gradient methods with space dilation in the direction of the difference between two successive gradients is investigated in connection with minimization problems for piecewise-smooth functions.
Abstract: The gradient method with space dilation in the direction of the difference between two successive gradients was proposed in [1]. Later in [2] isolated modifications o f this method were investigated experimentally in connection with minimization problems for smooth functions and in [3] and [5-7] for the solution o f minimization problems for piecewise-smooth functions. Up to the present time a large number o f complex practical problems have been solved by this method and one can speak of its effectiveness with great confidence, particularly for solving minimax problems and problems o f convex programming (with use of unsmooth penalty functions), and for minimizing functions of "ravine" type. However, the theoretical investigation of the convergence of this method was done only for one quite special case in [1] and [2]. In this paper the question o f convergence of a certain class o f monotonic gradient methods with space dilation in the direction o f the difference between two successive gradients is investigated in connection with minimization problems for piecewisesmooth functions.
TL;DR: In this paper, a general notion of positive and bounded variation is introduced for functions on a commutative semigroup with involution, and an integral representation for these functions is given.
Abstract: A general notion of positive and bounded variation is introduced for functions on a commutative semigroup with involution. An integral representation for these functions is given. Applications to specific semigroups provide solutions to moment problems over convex bodies in R as well as a recovery of the Bochner-Herglotz-Weil theorem for discrete groups.
TL;DR: In this article, a function is said to be convex if it is supported, at each point, by some member of a family of functions 9, and a function can be characterized and studied in terms of generalized convexity.
Abstract: With respect to a given family of functions 9, a function is said to be ^-convex, if it is supported, at each point, by some member of 9. For particular choices of 9 one obtains the convex functions and the generalized convex functions in the sense of Beckenbach. ^-convex functions are characterized and studied, retaining some essential results of classical convexity.
TL;DR: In this paper, a methodology to design a stormwater sewer system using a nonlinear programming approach is developed, which is divided in five steps: hydrology, set up of the technological constraints, optimization with the Rosen's projected gradient method with the pipe diameters considered as continuous variables, standardization of the diameters, and post-optimal analysis of the piezometric surface.
Abstract: A methodology to design a stormwater sewer system using a nonlinear programming approach is developed. It is divided in five steps: (1) hydrology, (2) set up of the technological constraints, (3) optimization with the Rosen's projected gradient method with the pipe diameters considered as continuous variables, (4) standardization of the diameters, and (5) post-optimal analysis of the piezometric surface. The cost function includes the purchase and installation and the excavation costs of every pipe. It is a convex programming problem; therefore the minimum solution is an absolute minimum. A subsequent paper to appear in this journal will illustrate the complete design procedure and the effect of certain parameters on the design.
TL;DR: In this paper, a convergent algorithm for mixed-integer convex programming problems with integer objective function is presented, which combines Kelley's cutting-plane provedure and Gomory's second algorithm.
Abstract: For the solution of mixed integer convex programming problems with integer objective function an algorithm is established, which combines Kelley's cutting-plane provedure and Gomory's second algorithm. Under the assumptions to be made for these procedures a convergent algorithm results by introducing an additional cut. In this algorithm the number of constraints in the linear subproblems is bounded. In case of pure integer problems the algorithm becomes finite.
TL;DR: In this article, the existence of a solution under feasibility assumptions to a convex minimization problem over polyhedral cones in complex space is given by using the fact that the problem of solving convex optimization programs naturally leads to the consideration of the nonlinear complementarity problem.
Abstract: A theorem on the existence of a solution under feasibility assumptions to a convex minimization problem over polyhedral cones in complex space is given by using the fact that the problem of solving a convex minimization program naturally leads to the consideration of the following nonlinear complementarity problem: given g : C n → C n , find z such that g ( z ) ∈ S *, z ∈ S , and Re 〈 g ( z ), z 〉 = 0, where S is a polyhedral cone and S * its polar.