The complementary unboundedness of dual feasible solution sets in convex programming

F.E. Clark has shown that if at least one of the feasible solution sets for a pair of dual linear programming problems is nonempty then at least one of them is both nonempty and unbounded. Subsequently, M. Avriel and A.C. Williams have obtained the same result in the more general context of (prototype posynomial) geometric programming. In this paper we show that the same result is actually false in the even more general context of convex programming — unless a certain regularity condition is satisfied.We also show that the regularity condition is so weak that it is automatically satisfied in linear programming (prototype posynomial) geometric programming, quadratic programming (with either linear or quadratic constraints),l _p -regression analysis, optimal location, roadway network analysis, and chemical equilibrium analysis. Moreover, we develop an equivalent regularity condition for each of the usual formulations of duality.Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-73-2516.     

Transcendental geometric programs

This paper treats a class of posynomial-like functions whose variables may appear also as exponents or in logarithms. It is shown that the resulting programs, called transcendental geometric programs, retain many useful properties of ordinary geometric programs, although the new class of problems need not have unique minima and cannot, in general, be transformed into convex programs. A duality theory, analogous to geometric programming duality, is formulated under somewhat more restrictive conditions. The dual constraints are not all linear, but the notion ofdegrees of difficulty is maintained in its geometric programming sense. One formulation of the dual program is shown to be a generalization of the chemical equilibrium problem where correction factors are added to account for nonideality. Some of the computational difficulties in solving transcendental programs are discussed briefly.This research was partially supported by the National Institute of Health Grant No. GM-14789; Office of Naval Research under Contract No. N00014-75-C-0276; National Science Foundation Grant No. MPS-71-03341 A03; and the US Atomic Energy Commission Contract No. AT(04-3)-326 PA #18.     

Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs

The paper concerns the study of new classes of parametric optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain, among other constraints, infinitely many inequality constraints. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We focus on DC infinite programs with objectives given as the difference of convex functions subject to convex inequality constraints. The main results establish efficient upper estimates of certain subdifferentials of (intrinsically nonsmooth) value functions in DC infinite programs based on advanced tools of variational analysis and generalized differentiation. The value/marginal functions and their subdifferential estimates play a crucial role in many aspects of parametric optimization including well-posedness and sensitivity. In this paper we apply the obtained subdifferential estimates to establishing verifiable conditions for the local Lipschitz continuity of the value functions and deriving necessary optimality conditions in parametric DC infinite programs and their remarkable specifications. Finally, we employ the value function approach and the established subdifferential estimates to the study of bilevel finite and infinite programs with convex data on both lower and upper level of hierarchical optimization. The results obtained in the paper are new not only for the classes of infinite programs under consideration but also for their semi-infinite counterparts.     

A computational study of methods for solving polynomial geometric programs

A computational comparison of several methods for dealing with polynomial geometric programs is presented. Specifically, we compare the complementary programs of Avriel and Williams (Ref. 1) with the reversed programs and the harmonic programs of Duffin and Peterson (Refs. 2, 3). These methods are used to generate a sequence of posynomial geometric programs which are solved using a dual algorithm.The authors would like to acknowledge the helpful comments of the referees. Also, they would like to acknowledge the programming assistance of Mr. S. N. Wong of The Pennsylvania State University. The first author's research was supported in part by a Research Initiation Grant awarded through The Pennsylvania State University.     

Fuzzy正项几何规划的直接算法

将Fuzzy正项几何规划化的一变量有上、下界限制的Fuzzy正项几何规划,利用Fuzzy几何不等式,又将该变量有上、下界限制的Fuzzy正项几何规划化为单项Fuzzy正项几何规划,提出基于Fuzzy值集割平面法的两种直接算法,并通过一个数值例证实该方法的有效性。     

Convergence of a Tuy-type algorithm for concave minimization subject to linear inequality constraints

A modification of Tuy's cone splitting algorithm for minimizing a concave function subject to linear inequality constraints is shown to be convergent by demonstrating that the limit of a sequence of constructed convex polytopes contains the feasible region. No geometric tolerance parameters are required.Research supported by National Science Foundation Grant ENG 76-12250     

MM algorithms for geometric and signomial programming

This paper derives new algorithms for signomial programming, a generalization of geometric programming. The algorithms are based on a generic principle for optimization called the MM algorithm. In this setting, one can apply the geometric-arithmetic mean inequality and a supporting hyperplane inequality to create a surrogate function with parameters separated. Thus, unconstrained signomial programming reduces to a sequence of one-dimensional minimization problems. Simple examples demonstrate that the MM algorithm derived can converge to a boundary point or to one point of a continuum of minimum points. Conditions under which the minimum point is unique or occurs in the interior of parameter space are proved for geometric programming. Convergence to an interior point occurs at a linear rate. Finally, the MM framework easily accommodates equality and inequality constraints of signomial type. For the most important special case, constrained quadratic programming, the MM algorithm involves very simple updates.     

Solving Posynomial Geometric Programming Problems via Generalized Linear Programming

This paper revisits an efficient procedure for solving posynomial geometric programming (GP) problems, which was initially developed by Avriel et al. The procedure, which used the concept of condensation, was embedded within an algorithm for the more general (signomial) GP problem. It is shown here that a computationally equivalent dual-based algorithm may be independently derived based on some more recent work where the GP primal-dual pair was reformulated as a set of inexact linear programs. The constraint structure of the reformulation provides insight into why the algorithm is successful in avoiding all of the computational problems traditionally associated with dual-based algorithms. Test results indicate that the algorithm can be used to successfully solve large-scale geometric programming problems on a desktop computer.     

An infeasible interior-point algorithm for solving primal and dual geometric programs

In this paper an algorithm is presented for solving the classical posynomial geometric programming dual pair of problems simultaneously. The approach is by means of a primal-dual infeasible algorithm developed simultaneously for (i) the dual geometric program after logarithmic transformation of its objective function, and (ii) its Lagrangian dual program. Under rather general assumptions, the mechanism defines a primal-dual infeasible path from a specially constructed, perturbed Karush-Kuhn-Tucker system.Subfeasible solutions, as described by Duffin in 1956, are generated for each program whose primal and dual objective function values converge to the respective primal and dual program values. The basic technique is one of a predictor-corrector type involving Newton’s method applied to the perturbed KKT system, coupled with effective techniques for choosing iterate directions and step lengths. We also discuss implementation issues and some sparse matrix factorizations that take advantage of the very special structure of the Hessian matrix of the logarithmically transformed dual objective function. Our computational results on 19 of the most challenging GP problems found in the literature are encouraging. The performance indicates that the algorithm is effective regardless of thedegree of difficulty, which is a generally accepted measure in geometric programming. Research supported in part by the University of Iowa Obermann Fellowship and by NSF Grant DDM-9207347.     

Relaxation methods for monotropic programs

We propose a dual descent method for the problem of minimizing a convex, possibly nondifferentiable, separable cost subject to linear constraints. The method has properties reminiscent of the Gauss-Seidel method in numerical analysis and uses the-complementary slackness mechanism introduced in Bertsekas, Hosein and Tseng (1987) to ensure finite convergence to near optimality. As special cases we obtain the methods in Bertsekas, Hosein and Tseng (1987) for network flow programs and the methods in Tseng and Bertsekas (1987) for linear programs.Work supported by the National Science Foundation under grant NSF-ECS-8519058 and by the Army Research Office under grant DAAL03-86-K-0171.This author is also supported by Canadian NSERC under Grant U0505.     

一般广义几何规划问题的一种有效数值方法

一般广义几何规划问题的一种有效数值方法张可村，杨波艇（西安交通大学）ＡＮＥＦＦＥＣＴＩＶＥＮＵＭＥＲＩＣＡＬＭＥＴＨＯＤＦＯＲＧＥＮＥＲＡＬＳＩＧＮＯＭＩＡＬＧＥＯＭＥＴＲＩＣＰＲＯＧＲＡＭＭＩＮＧＰＲＯＢＬＥＭＳ￥ＺｈａｎｇＫｅ－ｃｕｎ；ＹａｎｇＢ...     

Optimal design of pitched laminated wood beams

The optimal design of a pitched laminated wood beam is considered. An engineering formulation is given in which the volume of the beam is minimized. The problem is then reformulated and solved as a generalized geometric (signomial) program. Sample designs are presented.This research was partially supported by the Office of Naval Research under Contracts Nos. N00014-75-C-0267 and N00014-75-C-0865; by the US Energy Research and Development Administration Contract No. E(04-3)-326 PA-18; and by the National Science Foundation, Grant No. DCR75-04544 at Stanford University. This work was carried out during the first author's stay at the Management Science Division of the University of British Columbia and the Systems Optimization Laboratory of Stanford University. The authors are indebted to Mr. S. Liu and Mrs. M. Ratner for their assistance in performing the computations.     

A transformation technique for optimal control problems with partially linear state inequality constraints

This paper considers optimal control problems involving the minimization of a functional subject to differential constraints, terminal constraints, and a state inequality constraint. The state inequality constraint is of a special type, namely, it is linear in some or all of the components of the state vector.A transformation technique is introduced, by means of which the inequality-constrained problem is converted into an equality-constrained problem involving differential constraints, terminal constraints, and a control equality constraint. The transformation technique takes advantage of the partial linearity of the state inequality constraint so as to yield a transformed problem characterized by a new state vector of minimal size. This concept is important computationally, in that the computer time per iteration increases with the square of the dimension of the state vector.In order to illustrate the advantages of the new transformation technique, several numerical examples are solved by means of the sequential gradient-restoration algorithm for optimal control problems involving nondifferential constraints. The examples show the substantial savings in computer time for convergence, which are associated with the new transformation technique.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-76-3075, and by the National Science Foundation, Grant No. MCS-76-21657.     

Weak monotonicity inequality and partial regularity for harmonic maps

The notion of locally weak monotonicity inequality for weakly harmonic maps is introduced and various results on this class of maps are obtained. For example, the locally weak monotonicity inequality is nearly equivalent to theε-regularity. Project supported by the National Natural Science Foundation of China (Grant No. 19571028) and the Guangdong Provincial Natural Science Foundation of China.     

A superlinearly convergent method for minimization problems with linear inequality constraints

A method is described for minimizing a continuously differentiable functionF(x) ofn variables subject to linear inequality constraints. It can be applied under the same general assumptions as any method of feasible directions. IfF(x) is twice continuously differentiable and the Hessian matrix ofF(x) has certain properties, then the algorithm generates a sequence of points which converges superlinearly to the unique minimizer ofF(x). No computation of secondorder derivatives is required.Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462 and by the National Science Foundation under Research Grant GP.33033.     

Calculation of bounds on variables satisfying nonlinear inequality constraints

Existing techniques for solving nonconvex programming problems often rely on the availability of lower and upper bounds on the problem variables. This paper develops a method for obtaining these bounds when not all of them are availablea priori. The method is a generalization of the method of Fourier which finds bounds on variables satisfying linear inequality constraints. First, nonlinear inequality constraints are converted to equivalent sets of separable constraints. Generalized variable elimination techniques are used to reduce these to constraints in one variable. Bounds on that variable are obtained and an inductive process yields bounds on the others.Research Sponsored by the Office of Naval Research Grant N00014-89-J-1537.     

Convex analysis treated by linear programming

The theme of this paper is the application of linear analysis to simplify and extend convex analysis. The central problem treated is the standard convex program — minimize a convex function subject to inequality constraints on other convex functions. The present approach uses the support planes of the constraint region to transform the convex program into an equivalent linear program. Then the duality theory of infinite linear programming shows how to construct a new dual program of bilinear type. When this dual program is transformed back into the convex function formulation it concerns the minimax of an unconstrained Lagrange function. This result is somewhat similar to the Kuhn—Tucker theorem. However, no constraint qualifications are needed and yet perfect duality maintains between the primal and dual programs.Work prepared under Research Grant DA-AROD-31-124-71-G17, Army Research Office (Durham).     

Optimizing a polyhedral-semidefinite relaxation of completely positive programs

It has recently been shown (Burer, Math Program 120:479–495, 2009) that a large class of NP-hard nonconvex quadratic programs (NQPs) can be modeled as so-called completely positive programs, i.e., the minimization of a linear function over the convex cone of completely positive matrices subject to linear constraints. Such convex programs are NP-hard in general. A basic tractable relaxation is gotten by approximating the completely positive matrices with doubly nonnegative matrices, i.e., matrices which are both nonnegative and positive semidefinite, resulting in a doubly nonnegative program (DNP). Optimizing a DNP, while polynomial, is expensive in practice for interior-point methods. In this paper, we propose a practically efficient decomposition technique, which approximately solves the DNPs while simultaneously producing lower bounds on the original NQP. We illustrate the effectiveness of our approach for solving the basic relaxation of box-constrained NQPs (BoxQPs) and the quadratic assignment problem. For one quadratic assignment instance, a best-known lower bound is obtained. We also incorporate the lower bounds within a branch-and-bound scheme for solving BoxQPs and the quadratic multiple knapsack problem. In particular, to the best of our knowledge, the resulting algorithm for globally solving BoxQPs is the most efficient to date.     

Minimal covers,minimal sets and canonical facets of the posynomial knapsack polytope

An irredundant representation of the 0–1 solutions to a posynomial inequality in terms of covering constraints induced by minimal covers is given. This representation is further strengthened using extended covering constraints induced by maximal extensions of minimal covers. Necessary, sufficient, and in a special case necessary and sufficient conditions for an extended covering constraint induced by a minimal set to be a facet of the posynomial knapsack polytope are given.     

Sensitivity analysis in posynomial geometric programming

Sensitivity analysis results for general parametric posynomial geometric programs are obtained by utilizing recent results from nonlinear programming. Duality theory of geometric programming is exploited to relate the sensitivity results derived for primal and dual geometric programs. The computational aspects of sensitivity calculations are also considered.This work was part of the doctoral dissertation completed in the Department of Operations Research, George Washington University, Washington, DC. The author would like to express his gratitude to the thesis advisor, Prof. A. V. Fiacco, for overall guidance and stimulating discussions which inspired the development of this research work.