Deterministic global optimization with partition sets whose feasibility is not known: Application to concave minimization,reverse convex constraints,DC-programming,and Lipschitzian optimization

A crucial problem for many global optimization methods is how to handle partition sets whose feasibility is not known. This problem is solved for broad classes of feasible sets including convex sets, sets defined by finitely many convex and reverse convex constraints, and sets defined by Lipschitzian inequalities. Moreover, a fairly general theory of bounding is presented and applied to concave objective functions, to functions representable as differences of two convex functions, and to Lipschitzian functions. The resulting algorithms allow one to solve any global optimization problem whose objective function is of one of these forms and whose feasible set belongs to one of the above classes. In this way, several new fields of optimization are opened to the application of global methods.     

On Extensions of the Frank-Wolfe Theorems

In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the so-called Frank-Wolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasi-convex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty.     

Analytical Linear Inequality Systems and Optimization

In many interesting semi-infinite programming problems, all the constraints are linear inequalities whose coefficients are analytical functions of a one-dimensional parameter. This paper shows that significant geometrical information on the feasible set of these problems can be obtained directly from the given coefficient functions. One of these geometrical properties gives rise to a general purification scheme for linear semi-infinite programs equipped with so-called analytical constraint systems. It is also shown that the solution sets of such kind of consistent systems form a transition class between polyhedral convex sets and closed convex sets in the Euclidean space of the unknowns.     

Barrier method in nonsmooth convex optimization without convex representation

In this article we want to demonstrate that under mild conditions the barrier method is an effective solution approach for convex optimization problems whose objective is nonsmooth and whose feasible set is described by smooth inequality constraints in which all the constraint functions need not be convex.     

An interior semi-infinite programming method

In this paper, a new method for semi-infinite programming problems with convex constraints is presented. The method generates a sequence of feasible points whose cluster points are solutions of the original problem. No maximization over the index set is required. Some computational results are also presented.This work was partly supported by Republicka Zajednica za Nauku Socijalisticke Republike Srbije. The authors are indebted to Professor R. A. Tapia for encouraging the approach taken in this research.     

A graphical characterization of the efficient set for?convex multiobjective problems

In this paper, a graphical characterization, in the decision space, of the properly efficient solutions of a convex multiobjective problem is derived. This characterization takes into account the relative position of the gradients of the objective functions and the active constraints at the given feasible solution. The unconstrained case with two objective functions and with any number of functions and the general constrained case are studied separately. In some cases, these results can provide a visualization of the efficient set, for problems with two or three variables. Besides, a proper efficiency test for general convex multiobjective problems is derived, which consists of solving a single linear optimization problem.     

ε-Pareto Optimality Conditions for Convex Multiobjective Programming via Max Function

We consider a nondifferentiable convex multiobjective optimization problem whose feasible set is defined by affine equality constraints, convex inequality constraints, and an abstract convex set constraint. We obtain Fritz John and Kuhn–Tucker necessary and sufficient conditions for ε-Pareto optimality via a max function. We also provide some relations among ε-Pareto solutions for such a problem and approximate solutions for several associated scalar problems.     

Simplex-inspired algorithms for solving a class of convex programming problems

We consider a class of convex programming problems whose objective function is given as a linear function plus a convex function whose arguments are linear functions of the decision variables and whose feasible region is a polytope. We show that there exists an optimal solution to this class of problems on a face of the constraint polytope of dimension not more than the number of arguments of the convex function. Based on this result, we develop a method to solve this problem that is inspired by the simplex method for linear programming. It is shown that this method terminates in a finite number of iterations in the special case that the convex function has only a single argument. We then use this insight to develop a second algorithm that solves the problem in a finite number of iterations for an arbitrary number of arguments in the convex function. A computational study illustrates the efficiency of the algorithm and suggests that the average-case performance of these algorithms is a polynomial of low order in the number of decision variables. The work of T. C. Sharkey was supported by a National Science Foundation Graduate Research Fellowship. The work of H. E. Romeijn was supported by the National Science Foundation under Grant No. DMI-0355533.     

一类线性约束下不可微最优化问题的可行下降方法

本文给出了一类线性约束下不可微量优化问题的可行下降方法，这类问题的目标函数是凸函数和可微函数的合成函数，算法通过解系列二次规划寻找可行下降方向，新的迭代点由不精确线搜索产生，在较弱的条件下，我们证明了算法的全局收敛性     

Exterior Minimum-Penalty Path-Following Methods in Semidefinite Programming

A semidefinite programming problem is a mathematical program in which the objective function is linear in the unknowns and the constraint set is defined by a linear matrix inequality. This problem is nonlinear, nondifferentiable but convex. It covers several standard problems, such as linear and quadratic programming, and has many applications in engineering. In this paper, we introduce the notion of minimal-penalty path, which is defined as the collection of minimizers for a family of convex optimization problems, and propose two methods for solving the problem by following the minimal-penalty path from the exterior of the feasible set. Our first method, which is also a constraint-aggregation method, achieves the solution by solving a sequence of linear programs, but exhibits a zigzagging behavior around the minimal-penalty path. Our second method eliminates the above drawback by following efficiently the minimum-penalty path through the centering and ascending steps. The global convergence of the methods is proved and their performance is illustrated by means of an example.     

求解随机凸规划概率约束问题的对偶算法

1引言随机规划中的概率约束问题在工程和管理中有广泛的应用.因为问题中包含非线性的概率约束,它们的求解非常困难.如果目标函数是线性的,问题的求解就比较容易.给出了一个求解随机线性规划概率约束问题的综述.原-对偶算法和切平面算法是比较有效的.在本文中,我们讨论随机凸规划概率约束问题:     

一类线性乘积规划问题的分支定界缩减方法

提出了求解一类线性乘积规划问题的分支定界缩减方法, 并证明了算法的收敛性.在这个方法中, 利用两个变量乘积的凸包络技术, 给出了目标函数与约束函数中乘积的下界, 由此确定原问题的一个松弛凸规划, 从而找到原问题全局最优值的下界和可行解. 为了加快所提算法的收敛速度, 使用了超矩形的缩减策略. 数值结果表明所提出的算法是可行的.     

Descent methods for convex essentially smooth minimization

Consider the problem of minimizing a convex essentially smooth function over a polyhedral set. For the special case where the cost function is strictly convex, we propose a feasible descent method for this problem that chooses the descent directions from a finite set of vectors. When the polyhedral set is the nonnegative orthant or the entire space, this method reduces to a coordinate descent method which, when applied to certain dual of linearly constrained convex programs with strictly convex essentially smooth costs, contains as special cases a number of well-known dual methods for quadratic and entropy (either –logx orx logx) optimization. Moreover, convergence of these dual methods can be inferred from a general convergence result for the feasible descent method. When the cost function is not strictly convex, we propose an extension of the feasible descent method which makes descent along the elementary vectors of a certain subspace associated with the polyhedral set. The elementary vectors are not stored, but generated using the dual rectification algorithm of Rockafellar. By introducing an -complementary slackness mechanism, we show that this extended method terminates finitely with a solution whose cost is within an order of of the optimal cost. Because it uses the dual rectification algorithm, this method can exploit the combinatorial structure of the polyhedral set and is well suited for problems with a special (e.g., network) structure.This work was partially supported by the US Army Research Office Contract No. DAAL03-86-K-0171 and by the National Science Foundation Grant No. ECS-85-19058.     

METHOD OF CENTERS ALGORITHM FOR MULTI-OBJECTIVE PROGRAMMING PROBLEMS

In this paper, we consider a method of centers for solving multi-objective programming problems, where the objective functions involved are concave functions and the set of feasible points is convex. The algorithm is defined so that the sub-problems that must be solved during its execution may be solved by finite-step procedures. Conditions are given under which the algorithm generates sequences of feasible points and constraint multiplier vectors that have accumulation points satisfying the KKT conditions. Finally, we establish convergence of the proposed method of centers algorithm for solving multiobjective programming problems.     

A big-M type method for the computation of projections onto polyhedrons

In a previous work (Ref. 1), we examined some active set methods for the computation of the projection of a point onto a polyhedron when a feasible point is known. In this paper, we assume that such a point is not known and examine a method similar to the big-M method developed for the solution of linear programming problems. Special attention is given to the study of computing error propagation.This research was supported partially by the Progetto Finalizzato Informatica del CNR, Sottoprogetto P1, Sofmat.     

Nonsmooth semi-infinite programming problems with mixed constraints

We consider a nonsmooth semi-infinite programming problem with a feasible set defined by inequality and equality constraints and a set constraint. First, we study some alternative theorems which involve linear and sublinear functions and a convex set and we propose several generalizations of them. Then, alternative theorems are applied to obtain, under different constraint qualifications, several necessary optimality conditions in the type of Fritz-John and Karush-Kuhn-Tucker.     

An exact penalty function method with global convergence properties for nonlinear programming problems

In this paper a new continuously differentiable exact penalty function is introduced for the solution of nonlinear programming problems with compact feasible set. A distinguishing feature of the penalty function is that it is defined on a suitable bounded open set containing the feasible region and that it goes to infinity on the boundary of this set. This allows the construction of an implementable unconstrained minimization algorithm, whose global convergence towards Kuhn-Tucker points of the constrained problem can be established.     

Geodesic convexity in nonlinear optimization

The properties of geodesic convex functions defined on a connected RiemannianC ² k-manifold are investigated in order to extend some results of convex optimization problems to nonlinear ones, whose feasible region is given by equalities and by inequalities and is a subset of a nonlinear space.This research was supported in part by the Hungarian National Research Foundation, Grant No. OTKA-1044.     

A simplified test for optimality

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.This work was partly supported by NSF Grant No. Eng 76-10260.     

Contractibility of the Solution Sets in Strictly Quasiconcave Vector Maximization on Noncompact Domains

We study the contractibility of the efficient solution set of strictly quasiconcave vector maximization problems on (possibly) noncompact feasible domains. It is proved that the efficient solution set is contractible if at least one of the objective functions is strongly quasiconcave and any intersection of level sets of the objective functions is a compact (possibly empty) set. This theorem generalizes the main result of Benoist (Ref.1), which was established for problems on compact feasible domains.The authors thank Dr. T. D. Phuong, Dr. T. X. D. Ha, and the referees for helpful comments and suggestions.