Showing papers on "Convex optimization published in 1978"
TL;DR: In the general framework of inifinite-dimensional convex programming, two fundamental principles are demonstrated and used to derive several basic algorithms to solve a so-called "master" (constrained optimization) problem.
Abstract: In the general framework of inifinite-dimensional convex programming, two fundamental principles are demonstrated and used to derive several basic algorithms to solve a so-called "master" (constrained optimization) problem. These algorithms consist in solving an infinite sequence of "auxiliary" problems whose solutions converge to the master's optimal one. By making particular choices for the auxiliary problems, one can recover either classical algorithms (gradient, Newton-Raphson, Uzawa) or decomposition-coordination (two-level) algorithms. The advantages of the theory are that it clearly sets the connection between classical and two-level algorithms, It provides a framework for classifying the two-level algorithms, and it gives a systematic way of deriving new algorithms.
186 citations
TL;DR: In this article, the approximate optimal design problem is treated as a constrained convex programming problem, and a general class of optimal design algorithms is proposed from this point of view, and asymptotic convergence to optimal designs is also proved.
Abstract: The approximate optimal design problem is treated as a constrained convex programming problem. A general class of optimal design algorithms is proposed from this point of view. Asymptotic convergence to optimal designs is also proved. Related problems like the implementability problem for the infinite support case and the general step-length algorithms are discussed.
97 citations
44 citations
01 Jan 1978
TL;DR: In this paper, the Hestenes-Powell method of multipliers in convex programming is modified to obtain a superior global convergence property under a stopping criterion that is easier to implement.
Abstract: The Hestenes-Powell method of multipliers in convex programming is modified to obtain a superior global convergence property under a stopping criterion that is easier to implement. The convergence results are obtained from the theory of the proximal point algorithm for solving 0 ∈ T(z) when T is a maximal monotone operator. An extension is made to an algorithm for solving variational inequalities with explicit constraint functions.
40 citations
TL;DR: A simplification of recent characterizations of optimality in convex programming involving the cones of decrease and constancy of the objective and constraint functions is presented.
Abstract: A simplification of recent characterizations of optimality in convex programming involving the cones of decrease and constancy of the objective and constraint functions is presented. In the original characterization due to Ben-Israelet al., optimality was verified or a feasible direction of decrease was determined by considering a number of sets equal to the number of subsets of the set of binding constraints. By first finding the set of constraints which is binding at every feasible point, it is possible to verify optimality or determine a feasible direction of decrease by considering a single set. In the case of faithfully convex functions, this set can be found by solving at mostp systems of linear equations and inequalities, wherep is the number of constraints.
32 citations
TL;DR: A cutting hyperplane method is proposed, which successively considers the various cells of the decomposition, checks whether the cell contains an optimal solution to the problem, and imposes a convexity cut which rejects the whole cell from the feasibility region, which is shown to be finitely convergent.
Abstract: A piecewise convex program is a convex program such that the constraint set can be decomposed in a finite number of closed convex sets, called the cells of the decomposition, and such that on each of these cells the objective function can be described by a continuously differentiable convex function.
In a first part, a cutting hyperplane method is proposed, which successively considers the various cells of the decomposition, checks whether the cell contains an optimal solution to the problem, and, if not, imposes a convexity cut which rejects the whole cell from the feasibility region. This elimination, which is basically a dual decomposition method but with an efficient use of the specific structure of the problem is shown to be finitely convergent.
The second part of this paper is devoted to the study of some special cases of piecewise convex program and in particular the piecewise quadratic program having a polyhedral constraint set. Such a program arises naturally in stochastic quadratic programming with recourse, which is the subject of the last section.
24 citations
TL;DR: In this paper, a notion of conjugate is proposed for convex vector-valued functions, and the Fenchel's problem is solved in convex multiple objective optimization and the associated duality theorem is proved.
20 citations
TL;DR: In this paper, the optimal impulsive control of systems arising from linear compartment models for drug distribution in the human body is considered, where a system of linear, time-invariant, homogeneous differential equations is given along with a set of continuous constraints on state and control.
Abstract: The 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. 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 an ordinary finite-dimensional convex programming problem. An iterative algorithm is given for the situation in which the critical points cannot all be determineda priori.
19 citations
TL;DR: It is shown that the entire sequence of iterates generated by each of the two proposed algorithms converges to a minimum although the economic function is not strictly convex.
Abstract: We treat an extension of the generalized Fermat—Weber problem with convex cost functions. It is shown that the entire sequence of iterates (as opposed to selected subsequences) generated by each of the two proposed algorithms converges to a minimum although the economic function is not strictly convex. The general idea is to associate, with the economic function calledh, a family of more regular strictly convex functions, the lower envelope of which is the functionh.
18 citations
TL;DR: In this paper, a sufficient optimality criterion of the classical Kuhn-Tucker and Fritz John type without assuming differentiability of functions involved has been established in the nonlinear case.
17 citations
01 Jan 1978
TL;DR: A new algorithm is proposed which is an extension of the Cheney-Goldstein algorithm for solving the same problem but under much weaker assumptions: the sets S and U are arbitrary and the mappings blt.
Abstract: A gênerai exchange algorithm is gwen for the mimmizatwn ofa convex function with equahty and inequahty constraintsAt is a generahzation of the Cheney-Goldstein algorithm, but following an idea gwen by Topfer, afimte séquence ofsub-problems the dimension ofwhich is decreasing, is considered at each itération Gwen a positive number 8, under very gênerai conditions, it is proved that the method, after afimte number of itérations, leads to an "e-solutwn" In 1959, Cheney and Goldstein [6] (see also Goldstein [7]) proposed an algorithm for solving the problem of minimizing a convex function: « \ ( £ u / * j(x)=max( iJbl(t)xl — c\ teS \ i = l under the constramts: £ bt{t)xt^c(t) for ail tel/, i = i where S and U are two disjoint compact sets and blt. . ., bn, c are continuous real functions defîned on S u U. At each itération v of this algorithm, a polyhedral approximation of the problem is associated to a suitable subset A consisting of n + 1 points of S u U. Using the exchange theorem (Stiefel [11, 12, 13]; see also [8, 9]) a new element t e S u U is introduced: A x =(A\t0) u t\ We propose hère a new algorithm which is an extension of the CheneyGoldstein algorithm for solving the same problem but under much weaker assumptions: the sets S and U are arbitrary and the mappings blt. . ., bn)c are (*) Rtui décembre 1977 () Mathématiques appliquées I M A G Université Scientifique et Médicale de Grenoble () U E R de Sciences, Université de Saiat-Étienne, Saint-Étienne R A I R O Analyse numénque/Numencal Analysis, vol 12, n° 4, 1978 378 P. J. LAURENT, C. CARASSO only supposed to be bounded. Moreover, no Haar condition is introduced. At each itération, we consider a séquence of nested minimization problems. The algorithm is based on an extension of the exchange theorem in which the exchanged quantities are not just a single point {see [3, 4]). The idea of the algorithm is similar to the recursive method introduced by Töpfer [14], [15] {see also [3]) for problems of Tchebycheff best approximation. In the case of a best approximation problem the algorithm becomes an extension of the Rémès aigorithm {see [5]). For other applications, see [2]. 1. PROBLEM AND ASSUMPTIONS We dénote by E the n-dimensional Euclidean space and by < x, x' > the usual inner-product of x and x' in E. 1 .1 . The minimization problem We dénote by L a finite set with l éléments (/ < n) and by S and U two arbitrary sets. Suppose that L, S and U have no common point and let T = S u U. Let b and c be two bounded mappings from L u Tinto E and R respectively (i. e., b{T) and c{T) are bounded). We define the functionals ƒ and g by: / (x)=Sup«x,6( t )>-c( t ) ) . teS
TL;DR: In this paper, a mathematical model is considered to derive dual extremum principles and global error bounds for monotone hyperelastic boundary-value problems, and sufficient conditions for the convexity of the strain energy density are established.
Abstract: Using some concepts of convex analysis a mathematical model is considered to derive dual extremum principles and global error bounds for monotone hyperelastic boundary-value problems. Introducing the two vector spaces of displacement gradients and of Piola stresses, which can be put into duality by a bilinear form, conjugate, and complementary energy functions are defined with the aid of Fenchel's transformation. Sufficient conditions for the convexity of the strain energy density are established. If the strain energy density is a convex function in some convex subset of the vector space of displacement gradients, dual extremum principles can be obtained by using Fenchel's inequality. They provide global error bounds for the solution of hyperelastic boundary-value problems.
TL;DR: It is found that Clark's Theorem applies unchanged to dual convex programs.
Abstract: Given a linear minimization program, then there is an associated linear maximization program termed the dual. F. E. Clark proved the following theorem. “If the set of feasible points of one program is bounded, then the set of feasible points of the other program is unbounded.” A convex program is the minimization of a convex function subject to the constraint that a number of other convex functions be nonpositive. As is well known, a dual maximization problem can be defined in terms of the Lagrange function. The dual objection function is the infimum of the Lagrange function. The feasible Lagrange multipliers are those satisfying: (i) the multipliers are nonnegative and (ii) the dual objective function is not negative infinity. It is found that Clark's Theorem applies unchanged to dual convex programs. Moreover, the programs have equal values.
TL;DR: In this paper, a survey about the following directions of generalizations for separation theorems involving some new results are given: Separation of finite families of convex sets, separation of sets in product spaces, separation in projective spaces and separation in convexity spaces; as a basis for all these considerations vector spaces without topology are used.
Abstract: In this paper a survey about the following directions of generalizations for separation theorems involving some new results are given: Separation of finite families of convex sets; separation of sets in product spaces; separation of convex sets in projective spaces; separation of convex sets in convexity spaces. As a basis for all these considerations vector spaces without topology are used.
TL;DR: In this article, necessary and sufficient conditions are given for a Banach space to be given on equivalent norm under which it is δ-uniformly convex orρuniformally smooth, where δ andρ are suitable Orlicz functions.
Abstract: Necessary and sufficient conditions are given, in terms of the behaviour of martingales, for a Banach space to be given on equivalent norm under which it isδ-uniformly convex orρ-uniformly smooth, whereδ andρ are suitable Orlicz functions.
TL;DR: In this article, an algorithm was developed for finding the global minimum of a continuously differentiable function on a compact interval in R 1, where the function is assumed to be the sum of a convex and a concave function, each of which belongs to C 1 [a, b ].
TL;DR: In this paper, the authors reformulated three popular feasible direction methods for solving convex programming problems so that they now work in the absence of Slater's condition or any other constraint qualification.
Abstract: Three popular feasible direction methods for solving convex programming problems are reformulated so that they now work in the absence of Slater’s condition or any other constraint qualification.
01 Jan 1978
TL;DR: This paper describes an application of convex programming for optimal long term planning of an electrical system and its applications in medicine, engineering, and telecommunications.
Abstract: This paper describes an application of convex programming for optimal long term planning of an electrical system
TL;DR: In this article, a nonlinear feehack law of subgradient type is used to synthesize a convex optimal control with convex cost criterion and an infinite time horizon associated with linear evolution equations in Hilbert spaces.
TL;DR: In this article, the authors identify essentially the minimal conditions permitting the replacement of a convex minimization problem by an equivalent one, in which each of the original convex functions is replaced by a suitably chosen affine minorant.
Abstract: It is often possible to replace a convex minimization problem by an equivalent one, in which each of the original convex functions is replaced by a suitably chosen affine minorant. In this paper we identify essentially the minimal conditions permitting this replacement, and also shed light on the close and complete link between such optimal affine minorants and certain optimal dual vectors. An application to the ordinary convex programming problem is included.
TL;DR: An approach is presented for estimating the impulse response function of a multidegree-of-freedom structural system that discretizes the time axis into contiguous time segments and estimates the structure’s impulse Response function for each time segment.
Abstract: An approach is presented for estimating the impulse response function of a multidegree-of-freedom structural system. The approach discretizes the time axis into contiguous time segments and estimates the structure's impulse response function for each time segment. The identification problem is formulated in the time domain and in matrix form which is solvable using math programming techniques. The solution is a global optimum because the problem is a convex programming problem.
TL;DR: In this article, the problem of maximizing the ratio of a concave function to a convex function subject to constraints in terms of upper bounds on convex functions and with each variable occurring in a single constraint was considered.
Abstract: The problem considered is that of maximizing the ratio of a concave function to a convex function subject to constraints in terms of upper bounds on convex functions and with each variable occurring in a single constraint. It is demonstrated that the Kuhn-Tucker conditions are sufficient for a feasible solution to be optimal.
TL;DR: Improvements on existing convergence rate results are obtained for the important type of descent algorithm which, at each interation, uses a search direction making an angle with the negative gradient which is smaller than a prespecified angle.
Abstract: Convergence to the minimal value is studied for the important type of descent algorithm which, at each interation, uses a search direction making an angle with the negative gradient which is smaller than a prespecified angle. Improvements on existing convergence rate results are obtained.
01 Jan 1978
TL;DR: This exposition presents a state-of-the-art survey of models and algorithms for the convex cost network flow problem.
Abstract: : This exposition presents a state-of-the-art survey of models and algorithms for the convex cost network flow problem. (Author)
TL;DR: A treatment of dual projection and restriction methods developed in terms of dual generalized geometric programming problems is given in this article. But it is only in the original polynomial form and not in the generalized form.
Abstract: Mathematical programming problems with unattained infima or unbounded optimal solution sets are dual to problems which lack interior points, e.g., problems for which the Slater condition fails to hold or for which the hypothesis of Fenchel’s theorem fails to hold. In such cases, it is possible to project the unbounded problem onto a subspace and to restrict the dual problem to an affine set so that the infima are not altered. After a finite sequence of such projections and restrictions, dual problems are obtained which have bounded optimal solution sets and interior points. Although results of this kind have occasionally been used in other contexts, it is in geometric programming (both in the original polynomial form and the generalized form) where such methods appear most useful. In this paper, we present a treatment of dual projection and restriction methods developed in terms of dual generalized geometric programming problems. Analogous results are given for Fenchel and ordinary dual problems.
01 Jan 1978
TL;DR: In this paper, the authors generalize the usual separation "theorems" to separation involving (continuous) linear operators mapping a (topological) vector space in a (normal topological) partially ordered vector space.
Abstract: In this paper we generalize the usual separation "theorems (where the separation is carried out by (continuous) linear functionals) to separation involving (continuous) linear operators mapping a (topological) vector space in a (normal topological) partially ordered vector space. Key words; Separation theorems, convex sets. AMSj 46-00, 52A05 Hef. 2.: 7.97 (46A40) §
TL;DR: In this paper, a class of penalty functions for solving convex programming problems with general constraint sets is considered and convergence theorems for penalty methods are established by utilizing the concept of infimal convergence of a sequence of functions.
Abstract: A class of penalty functions for solving convex programming problems with general constraint sets is considered. Convergence theorems for penalty methods are established by utilizing the concept of infimal convergence of a sequence of functions. It is shown that most existing penalty functions are included in our class of penalty functions.