Showing papers on "Convex optimization published in 1980"
TL;DR: This paper derives several basic or abstract algorithms, already given in Ref. 1, and studies their convergence properties in the framework of i infinite-dimensional convex programming.
Abstract: The auxiliary problem principle allows one to find the solution of a problem (minimization problem, saddle-point problem, etc.) by solving a sequence of auxiliary problems. There is a wide range of possible choices for these problems, so that one can give special features to them in order to make them easier to solve. We introduced this principle in Ref. 1 and showed its relevance to decomposing a problem into subproblems and to coordinating the subproblems. Here, we derive several basic or abstract algorithms, already given in Ref. 1, and we study their convergence properties in the framework of i infinite-dimensional convex programming.
385 citations
TL;DR: A classification of convex NP -optimization problems is introduced and is applied to study the combinatorial structure of several optimization problems associated to well-known NP -complete sets and it is shown that structurally isomorphic problems have similar approximability properties.
249 citations
TL;DR: In this article, new classes of generalized convex functions are introduced, extending the concepts of quasi-convexity, pseudoconvexness, and their associate subclasses, and functions belonging to these classes satisfy certain local-global minimum properties.
Abstract: In this paper, new classes of generalized convex functions are introduced, extending the concepts of quasi-convexity, pseudoconvexity, and their associate subclasses. Functions belonging to these classes satisfy certain local-global minimum properties. Conversely, it is shown that, under some mild regularity conditions, functions for which the local-global minimum properties hold must belong to one of the classes of functions introduced.
75 citations
TL;DR: In this paper, a class of boundary-distributed linear control systems in Banach spaces is studied and a maximum principle for a convex control problem associated with such systems is obtained.
Abstract: A class of boundary-distributed linear control systems in Banach spaces is studied. A maximum principle for a convex control problem associated with such systems is obtained.
48 citations
TL;DR: In this article, the ordinary concept of supremum was generalized and duality theorems concerning convex programming problems were generalized, after which duality theorem was generalized for convex programs.
Abstract: In this paper the ordinary concept of supremum will be generalized. After wo have done so duality theorems concerning convex programming problems will be generalized.
45 citations
TL;DR: In this article, the question of whether a set is reachable by a nonlinear control system is answered in terms of the properties of a convex optimization problem, i.e., whether the value of the optimization problem is zero or infinity.
Abstract: The question of whether a set is reachable by a nonlinear control system is answered in terms of the properties of a convex optimization problem. The set is reachable or not according to whether the value of the optimization problem is zero or infinity. Our findings strengthen earlier sufficient conditions for a point not to be reachable, given in terms of Lyapunov-like functions, in that we assure that the functions exist. Our approach is to embed admissible trajectories in a space of measures, and to apply recently obtained results on the properties of measures arising in this way.
37 citations
Dissertation•
01 Jan 1980
TL;DR: The proposed JFCR problem is shown to be equivalent to a minimum delay routing problem corresponding to a bigger network and any minimumdelay routing algorithm can be converted into aJFCR algorithm.
Abstract: : A joint flow control and routing (JFCR) strategy is proposed for store and forward communication networks. The strategy is based on a convex optimization problem in terms of the average input rates and multi-commodity flows and is shown to have the following properties: First the average load of each buffer stays below some arbitrarily chosen level for the input rate and routing assignments of the strategy. This level can be chosen so as to upper bound the probability of buffer overflow arbitrarily. Secondly, by proper selection of the cost function, it is possible to utilize the network fully and to achieve a variety of different types of priorities in the services offered to the users. Finally, the routing assignments of the strategy correspond to a routing strategy/which tends to minimize the total delay when the network is lightly loaded and tends to prevent congestion when it is heavily loaded. Furthermore, the proposed JFCR problem is shown to be equivalent to a minimum delay routing problem corresponding to a bigger network. Accordingly, any minimum delay routing algorithm can be converted into a JFCR algorithm. Using this approach, a class of JFCR algorithms with distributed computations at the nodes are developed. Under certain conditions, a one to one correspondence is shown to exist in a store and forward network between the set of average input rates and the set of average number of outstanding packets of commodities. This unique correspondence is used to show that in practice the average input rates can be adjusted as desired by restricting the number of outstanding packets on each commodity (window strategy).
36 citations
TL;DR: In this article, it was shown that these measures are just unit mixtures of control-trajectory pairs and that admitting them does not alter the minimum value of the control problems.
33 citations
TL;DR: A method is presented for obtaining minimum discrimination information estimates of probability distributions using an extremal principle of Charnes and Cooper (1974) and, viewing M.D.I, estimation in a dual convex programming framework.
Abstract: A method is presented for obtaining minimum discrimination information (M.D.I.) estimates of probability distributions. This involves using an extremal principle of Charnes and Cooper (1974) and, viewing M.D.I, estimation in a dual convex programming framework. The resulting dual convex program is unconstrained and involves only exponential and linear terms, and hence is easily
33 citations
TL;DR: The cones of directions of constancy are used to derive: new as well as known optimality conditions; weakest constraint qualifications; and regularization techniques, for the convex programming problem.
Abstract: The cones of directions of constancy are used to derive: new as well as known optimality conditions; weakest constraint qualifications; and regularization techniques, for the convex programming problem In addition, the “badly behaved set” of constraints, ie the set of constraints which causes problems in the Kuhn—Tucker theory, is isolated and a computational procedure for checking whether a feasible point is regular or not is presented
TL;DR: In this article, an infinite dimensional mathematical programming problem is studied, which concerns the moment of collapse and the collapse state itself, of a plastic structure subjected to increasing loads, and the approximation of the infinite problem by a family of finite convex programming problems.
Abstract: We study an infinite dimensional mathematical programming problem, which arises naturally in solid mechanics. It concerns the moment of collapse, and the collapse state itself, of a plastic structure subjected to increasing loads. The duality between the «static» and «kinematic» theorems of limit analysis is well-known in discrete plasticity; we prove the same duality for a continuum including existence of the collapse fields for stresses and velocities as the primal and dual solutions. We then discuss the approximation of the infinite problem by a family of finite convex programming problems. Numerical results for classical problems in limit analysis where this discretization is based on finite elements will be published separately.
TL;DR: In this article, the authors extend the theory of generalized geometric programming to infinite dimensions in order to derive a dual problem for the convex optimal control problem, which transfers explicit constraints in the primal problem to the dual objective functional.
Abstract: The interest in convexity in optimal control and the calculus of variations has gone through a revival in the past decade. In this paper, we extend the theory of generalized geometric programming to infinite dimensions in order to derive a dual problem for the convex optimal control problem. This approach transfers explicit constraints in the primal problem to the dual objective functional.
TL;DR: Conditions for optimality which are both necessary and sufficient for convex programming in Banach spaces are given in this paper under a rather weak geometrical assumption on the existence of a relative radial point in the feasible set.
Abstract: Conditions for optimality which are both necessary and sufficient are given for convex programming in Banach spaces. They are derived under a rather weak geometrical assumption on the existence of a relative radial point in the feasible set. This assumption is superfluous in finite dimensions. Applications include a duality theorem without a constraint qualification, and a necessary and sufficient version of Pontryagin's principle for optimal control of a convex system.
TL;DR: A definition is proposed for the concept of shadow prices in nonconvex programming, for a nonlinear program with equality and inequality constraints, and results coincide with those already known on this subject in convex programming.
Abstract: In this paper a definition is proposed for the concept of shadow prices in nonconvex programming. For a nonlinear program with equality and inequality constraints, existence of these prices and bounds for their possible values are obtained under the Mangasarian—Fromowitz regularity condition. Their exact values and some continuity properties are obtained under the more restrictive linear independence regularity condition. A definition of equilibrium prices is also proposed. Under convexity assumptions, all definitions and results coincide with those already known on this subject in convex programming.
TL;DR: Quadratic functions are shown to satisfy sufficient conditions under which a function either attains its infimum on a nonempty polyhedral convex set or is unbounded below on some halfline of that set.
Abstract: The Frank—Wolfe theorem states that a quadratic function, bounded below on a nonempty polyhedral convex set, attains its infimum there. This paper gives sufficient conditions under which a function either attains its infimum on a nonempty polyhedral convex set or is unbounded below on some halfline of that set. Quadratic functions are shown to satisfy these sufficient conditions.
TL;DR: This work considers the linear estimation problem on a binary computer where the estimation parameters are constrained to be powers of two and thus all multiplications are replaced by shifts, and shows that the multiplication-free constraint does not generally increase the mean-square error significantly compared with the classical optimal solution.
Abstract: An optimal linear mean-square estimation algorithm is derived under the constraint that the algorithm be multiplication-free A classical linear estimation problem with block length N generally requires N^{2} multiplications For many on-line signal processing situations a large number of multiplications is objectionable This class of estimation problems includes the classical linear filtering of a random signal in random noise, as well as the linear equalization of digital data over a dispersive channel with additive noise Here we consider the linear estimation problem on a binary computer where the estimation parameters are constrained to be powers of two and thus all multiplications are replaced by shifts Then the optimal constrained linear estimation problem resembles an integer-programming problem except that the allowable discrete points are nonintegers The branch-and-bound principle is used to convert this minimization problem to a series of convex programming problems An algorithm is given for the solution as well as numerical results for filtering and data equalization These examples show that the multiplication-free constraint does not generally increase the mean-square error significantly compared with the classical optimal solution Furthermore, the intuitive "round to the nearest power of two" procedure for the estimation parameters can be inferior to the optimal brunch-and-bound solution
TL;DR: For a proper lower semi-continuous convex functional ƒ on a locally convex space E and a bounded subset G of E, a formula for sup ǫ(G) which is symmetric to the Lagrange multiplier theorem for convex minimization, obtained in this article, was shown.
Abstract: We prove, for a proper lower semi-continuous convex functional ƒ on a locally convex space E and a bounded subset G of E, a formula for sup ƒ(G) which is symmetric to the Lagrange multiplier theorem for convex minimization, obtained in [7], with the difference that for sup ƒ(G) Lagrange multiplier functionals need not exist. When ƒ is also continuous we give some necessary conditions for g0 ∈ G to satisfy ƒ(g0) = sup ƒ(G). Also, we give some applications to deviations and farthest points. Finally, we show the connections with the “hyperplane theorems” of our previous paper [8].
TL;DR: In this article, a duality theorem for the minimum of an arbitrary family of convex programs is given, which provides a single concave dual maximization and generalizes recent work in linear disjunctive programming.
Abstract: We produce a duality theorem for the minimum of an arbitrary family of convex programs This duality theorem provides a single concave dual maximization and generalizes recent work in linear disjunctive programming Homogeneous and symmetric formulations are studied in some detail, and a number of convex and nonconvex applications are given
TL;DR: In this paper, the collapse problem for plate bending is considered as an infinite dimensional mathematical programming problem, and the duality between the static and kinematic formulations of limit analysis is proved.
Abstract: The collapse problem for plate bending is considered as an infinite dimensional mathematical programming problem. The duality between the static and kinematic formulations of limit analysis is proved, and it is shown that limit fields for bending moments and displacement rates exist. Finally we analyze the approximation of the continuous problem by finite-dimensional convex programming problems using the finite element method.
TL;DR: In this paper, the Euler equation for the Lagrangian is formulated as a differential equation of second order in time, where V is a convex functional function and V is concave with respect to x.
Abstract: Many problems in mathematical physics can be formulated as differential equations of second order in time: with V a convex functional. This is the Euler equation for the Lagrangian which is convex with respect to x, and concave with respect to x.
TL;DR: In this article, the validity of the global pointwise maximum principle for convex optimal control problems with mixed control-phase variable inequality constraints was proved for a class of convex optimization problems.
Abstract: The validity of a global pointwise maximum principle is proved for a class of convex optimal control problems with mixed control-phase variable inequality constraints. No compatibility hypotheses are required, and singular multipliers are avoided.
TL;DR: Convergence rate information is obtained, which is sufficient to enable deduction of the number of iterations needed to achieve a specified reduction in the distance from the optimum (measured in terms of the cost).
Abstract: At each iteration, the algorithm determines a feasible descent direction by minimizing a linear or quadratic approximation to the cost on the feasible set. The algorithm is easy to implement if the approximation is easy to minimize on the feasible set, which happens in some important cases. Convergence rate information is obtained, which is sufficient to enable deduction of the number of iterations needed to achieve a specified reduction in the distance from the optimum (measured in terms of the cost). Existing convergence rates for algorithms for solving such convex problems are either asymptotic (and so do not enable the required number of iterations to be deduced) or decrease as the number of constraints increases. The convergence rate information obtained here, however, is independent of the number of constraints. For the case where the quadratic approximation to the cost is not strictly convex (which includes the linear approximation case), the diameter is the only property of the feasible set which affects the convergence rate information. If the quadratic approximation is strictly convex, the convergence rate is independent of the size and geometry of the feasible set. An application to a control-constrained optimal control problem is outlined.
Posted Content•
01 Jan 1980TL;DR: Two solution algorithms belonging to the class of feasible descent direction methods are proposed for solving the problem of predicting residential location, modal choice, and transportation system performance as a convex programming problem.
Abstract: A model is specified for predicting residential location, modal choice, and transportation system performance which is formulated as a convex programming problem. Two solution algorithms belonging to the class of feasible descent direction methods are proposed for solving this problem. Several extensions of the basic model are discussed and the associated solution algorithms are outlined. (This abstract was borrowed from another version of this item.)
TL;DR: In this paper, the equivalence of optimality over plans and optimality of a two-stage procedure related to dynamic programming was shown for completely convex stochastic programming problems.
TL;DR: A nonlinear convex programming problem is solved by methods of interval arithmetic which take into account the input errors and the round-off errors to find the solution of a nonlinear parameter dependent system of equations.
Abstract: A nonlinear convex programming problem is solved by methods of interval arithmetic which take into account the input errors and the round-off errors. The problem is reduced to the solution of a nonlinear parameter dependent system of equations. Moreover error estimations are developed for special problems with uniformly convex cost functions.
TL;DR: In this paper, the main characterization of Banach linear systems was extended to the case of locally convex spaces and the norm of the norm was replaced by an arbitrary continuous convex functional on the space.
Abstract: Using the subdifferential, we extend the main characterization of Banach linear systems
$$(F\mathop \to \limits^u X)$$
satisfying the Pontryagin maximum principle, given in our previous paper (Ref. 1), to the case whenF andX are locally convex spaces and the norm ofF is replaced by an arbitrary continuous convex functionalh onF.