Showing papers on "Convex optimization published in 1979"

Structural weight optimization by dual methods of convex programming

Abstract: This paper is mainly concerned with a new structural optimization method based upon the concept of duality in convex programming. This rigorous formulation permits justification of many intuitive procedures which are used in the classical optimality criteria approaches. Furthermore, the dual algorithms proposed in this paper do not suffer from the drawbacks inherent to the optimality criteria approach. The selection of the set of active constraints does not introduce any difficulty and is achieved correctly in all cases. The subdivision of the design variables in an active and passive group is intrinsically contained in the dual formulation. The efficiency of the dual algorithms is shown with reference to some problems for which the classical methods do not lead to satisfactory results.

Two Algorithms for Determining Volumes of Convex Polyhedra

Abstract: Determining volumes of convex n-dimensional polyhedra defined by a linear system of inequalities is useful in program analysis Two methods for computing these volumes are proposed (1) summing the volumes of stmphces which form the polyhedron, and (2) summing the volumes of (increasingly smaller) paralleleplpeds which can be fit into the polyhedron Assuming that roundoff errors are small, the first method is analytically exact whereas the second one converges to the exact solution at the expense of addmonal computer time Examples of polyhedra whose volumes were computed by programs representing the algorithms are also provided

Modified Lagrangians in convex programming and their generalizations

Abstract: In this paper a rather general class of modified Lagrangians is described for which the main results of the duality theory hold. Within this class two families of modified Lagrangians are taken into special consideration. The elements of the first family are characterized by so-called stability of saddle points and the elements of the second family generate smooth dual problems. The computational methods naturally connected with each of these two families are examined. Further a more general scheme is considered which exploits the idea of modification with respect to the problem of finding a root of a monotone operator. This scheme yields a unified approach to convex programming problems and to determination of saddle and equilibrium points as well as expands the class of modified Lagrangians.

Two-Segment Separable Programming

Abstract: New iterative separable programming techniques based on two-segment, piecewise-linear approximations are described for the minimization of convex separable functions over convex sets. These techniques have two advantages over traditional separable programming methods. The first is that they do not require the cumbersome “fine grid” approximations employed to achieve high accuracy in the usual separable programming approach. In addition, the new methods yield feasible solutions with objective values guaranteed to be within any specified tolerance of optimality. In computational tests with real-world problems of up to 500 “nonlinear” variables the approach has exhibited rapid convergence and yielded very close bounds on the optimal value.

On the Existence of Equilibria in Asymmetrical Multiclass-User Transportation Networks

Abstract: When studying multiclass-user transportation networks Dafermos made a strong assumption on symmetry in order to prove existence of a user-optimizing flow pattern. In this paper existence is proved without the symmetry-hypothesis and it is shown that the methods of convex programming are not applicable.

Convex/stochastic programming and multilocation inventory problems

Abstract: This paper examines a convex programming problem that arises in several contexts. In particular, the formulation was motivated by a generalization of the stochastic multilocation problem of inventory theory. The formulation also subsumes some “active” models of stochastic programming. A qualititative analysis of the problem is presented and it is shown that optimal policies have a certain geometric form. Properties of the optimal policy and of the optimal value function are described.

Necessary and sufficient conditions for local and global nondominated solutions in decision problems with multi-objectives

Abstract: In this paper, the decision problem with multi-objectives is considered, and the nondominated solutions associated with a polyhedral domination cone are discussed. The necessary and sufficient conditions for the solutions are given in the decision space rather than the objective space. The similarity of the solution conditions obtained in this article to the Kuhn-Tucker condition of a convex programming problem is examined. As an application of the solution condition, an algorithm to locate the set of all nondominated solutions is shown for the linear multi-objective decision problem.

On the lower semicontinuity of optimal sets in convex parametric optimization

Abstract: Regarding a special class of convex parametric problems sufficient conditions for the lower semicontinuity of the optimal solution sets are developed.

Technical Note—On the Estimation of Convex Functions

Abstract: We consider the estimation of a convex or concave relationship from a set of limited observations without prior specification of a functional form. A concave programming problem is shown to provide a “best” estimate for an arbitrary norm and n independent variables. The problem is shown to be well suited to a solution using the computational strategy of relaxation (a variant of generalized programming). An example illustrates the procedure and demonstrates the relationship to a procedure for n = 1 suggested by Dent.

Some new applications of the Fenchel-Rockafellar duality theorem: Lagrange multiplier theorems and hyperplane theorems for convex optimization and best approximation

Abstract: XEM xsM (the extension to complex scalars can be obtained with the usual methods-see [3,4]) and therefore, from now on, we shall omit the adjective “real” In several works it has been observed that Theorem A yields, as a particular case, the value of the intimum of a continuous convex functional f on a convex subset G of E, by taking in (11) h = -6,, where 6, is the indicator of the set G, ie,

Integral equations for the analytic extrapolation of scattering amplitudes with positivity constraints

Abstract: We investigate the stable analytic extrapolation of the scattering amplitudes with positive imaginary parts from the whole boundary or from a part of it to interior points. By using the Lagrange technique in convex optimization, we derive a system of singular integral equations which completely solve the problem. In the case of the extrapolation from the whole boundary this system reduces to the Fredholm equation recently obtained by Aubersonet al., using different considerations, in connection with a related extremum problem. Our results give also the correct analytic-extrapolation formulae in the limiting case of scattering data which are exactly the restriction to the boundary of some analytic function.

Characterizations of optimality in convex programming: the nondifferentiable case

Abstract: Primal, dual and saddle-point characterizations of optimality are given for convex programming in the general case (nondifferentiable functions and no constraint qualification).

Algorithmic equivalence in quadratic programming, I: A least-distance programming problem

Abstract: It is demonstrated that Wolfe's algorithm for finding the point of smallest Euclidean norm in a given convex polytope generates the same sequence of feasible points as does the van de Panne-Whinstonsymmetric algorithm applied to the associated quadratic programming problem Furthermore, it is shown how the latter algorithm may be simplified for application to problems of this type

Sufficient Conditions for Kuhn–Tucker Vectors in Convex Programming

Abstract: In this paper, we give new sufficient conditions for the existence of a Kuhn–Tucker vector for convex programs. These conditions generalize all the previously known ones.

Optimality conditions for a cone-convex programming problem

Abstract: Optimality conditions without constraint qualifications are given for the convex programming problem: Maximize f(x) such that g(x) ∈ B, where f maps X into R and is concave, g maps X into Rm and is B-concave, X is a locally convex topological vector space and B is a closed convex cone containing no line. In the case when B is the nonnegative orthant, the results reduce to some of those obtained recently by Ben-Israel, Ben-Tal and Zlobec.

Generalized isotone optimization with applications to starshaped functions

Abstract: This article considers a curve-fitting problem involving the minimization of the distance from a functionf to a convex cone of functions. A weighted uniform norm is considered as a measure of the distance. The domain of the functions is a partially ordered set, and the convex cone is defined by the isotonicity and nonnegativity conditions on functions. The problem has a linear programming formulation; however, explicit expressions for the optimal solutions have been obtained directly, thereby eliminating the necessity of using linear programming techniques. The results are applied to approximation by starshaped functions.

Symmetrized separable convex programming

Abstract: : The duality model for convex programming is analyzed from the viewpoint of perturbational duality theory. Relationships with the traditional Lagrangian model for ordinary programming are explored in detail, with particular emphasis placed on the respective dual problems, Kuhn-Tucker vectors, and extremality conditions. The case of homogeneous constraints is discussed by way of illustration. The Slater existence criterion for optimal Lagrange multipliers in ordinary programming is sharpened for the case in which some of the functions are polyhedral. The analysis generally covers nonclosed functions on general spaces and includes refinements to exploit polyhedrality in the finite-dimensional case. Underlying the whole development are basic technical facts which are developed concerning the Fenchel conjugate and preconjugate of the indicator function of an epigraph set.

Verallgemeinert konvexe vektorfunktionen und der anwendung in der vektoroptimierung

Abstract: Generalized convexity properties of vector functions in Rn /Rm with respect to a convex cone are studied. It is shown that the well-known connections between the various kinds of convexity, quasi-and pseudoconvexity of functionals can be extended to vector functions only to a part. The study of convexity properties with respect to convex polyhedral cones yields correspondences between the properties of vector functions and their components, respectively linear combinations of their components. As an example for the application of generalized convex vector functions a vector optimization problem with semi orderings defined by convex cones is considered. Besides a necessary and sufficient condition for the convexity of the feasible set, sufficient optimality conditions of Kuhn-Tucker and Fritz-John type are given.

A note on the continuity of the solution set of special dual optimization problems

Abstract: In this paper dual programs of convex optimization problems having a parametric objective function and a fixed linear feasible set are studied. By using some properties of the primal problem the continuity of the dual optimal solution set is proved.

Expansions of Generalized Completely Convex Functions

Abstract: The concept of a generalized completely convex function is extended arid a unified presentation is developed for expanding such functions by Taylor–Lidstone series. It is shown that these expansions are in fact tantamount to representation theorems for the elements of the cone of generalized completely convex functions in terms of the extreme rays.

Nonlinear programming and an optimization problem on infinitely differentiable functions

Abstract: A nonnegative, infinitely differentiable function o defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and ∫ 0 1 o(t)dt=1. In this article the following problem is considered. Determine Δ k =inf∫ 0 1 ∣vo(k)(t)∣dt, k=1,..., where o(k) denotes thekth derivative of o and the infimum is taken over the set of all mollifier functions. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. In this article, the structure of the problem of determining Δ k is analyzed, and it is shown that the problem is reducible to a nonlinear programming problem involving the minimization of a strictly convex function of [(k−1)/2] variables, subject to a simple ordering restriction on the variables. An optimization problem on the functions of bounded variation, which is equivalent to the nonlinear programming problem, is also developed. The results of this article and those from approximation of functions theory are applied elsewhere to derive numerical values of various mathematical quantities involved in this article, e.g., Δ k =k~22k−1 for allk=1, 2, ..., and to establish certain inequalities of independent interest. This article concentrates on problem reduction and equivalence, and not numerical value.

A general extremum principle for optimization problems

Abstract: THE FAMOUS Pontryagin [l] maximum principle has initiated a new trend in the optimization theory. Shortly after Dubovitskij and Milyutin [2] made an attempt to give a unified general approach to optimization problems by using an intersection property of convex cones. Later Halkin and Neustadt [3] proved a very general maximum principle. Recently Gahler [4] generalized the Dubovitskij-Milyutin principle. In doing so he confronted considerable restrictions concerning the involved topological vector spaces. However, the application of convex cones is of limited use as far as practical optimization problems are concerned. The main difficulty seems to be caused by the presence of equality constraints. In this respect, Lusternik’s [S] theorem on existence of tangent directions in Banach space is rather a major step in the right direction and also permits further generalizations. The Halkin-Neustadt principle seems to be limited in scope because it depends on Brouwer’s fixed point theorem. This paper presents a very general extremum principle which is closely related to [6]. The main feature of this principle is that it has two forms (see [6]). The primary form is rather constructive and can be used to find approximate solutions of the general optimization problem. In fact, in some special cases it reduces to the well-known method of feasible directions. The dual form leads to the Euler-Lagrange equation which is well known in classical cases. Another advantage of the principle is that it includes and extends all the most important necessary conditions for optimization problems.