Convexification, Concavification and Monotonization in Global Optimization

We show in this paper that via certain convexification, concavification and monotonization schemes a nonconvex optimization problem over a simplex can be always converted into an equivalent better-structured nonconvex optimization problem, e.g., a concave optimization problem or a D.C. programming problem, thus facilitating the search of a global optimum by using the existing methods in concave minimization and D.C. programming. We first prove that a monotone optimization problem (with a monotone objective function and monotone constraints) can be transformed into a concave minimization problem over a convex set or a D.C. programming problem via pth power transformation. We then prove that a class of nonconvex minimization problems can be always reduced to a monotone optimization problem, thus a concave minimization problem or a D.C. programming problem.     

On the convexification of nonlinear programming problems: An applications-oriented survey

Some important classes of decision models give rise to nonconvex minimization problems that, by domain or range transformation, are transformable into convex problems. Thus the powerful theoretical results and the efficiency of algorithms in convex programming can be exploited for a wide range of problems. If no convexifying transformation is at hand, one often requires to approximate a nonconvex objective by a convex one. Then a priori as well as post-computational errorbounds are of interest. The purpose of this paper is to outline briefly some of the ideas and results on convexification that may be useful in practice.     

Global minimization of a generalized convex multiplicative function

This paper discusses an algorithm for generalized convex multiplicative programming problems, a special class of nonconvex minimization problems in which the objective function is expressed as a sum ofp products of two convex functions. It is shown that this problem can be reduced to a concave minimization problem with only 2p variables. An outer approximation algorithm is proposed for solving the resulting problem.     

A generalized duality and applications

The aim of this paper is to present a nonconvex duality with a zero gap and its connection with convex duality. Since a convex program can be regarded as a particular case of convex maximization over a convex set, a nonconvex duality can be regarded as a generalization of convex duality. The generalized duality can be obtained on the basis of convex duality and minimax theorems. The duality with a zero gap can be extended to a more general nonconvex problems such as a quasiconvex maximization over a general nonconvex set or a general minimization over the complement of a convex set. Several applications are given.On leave from the Institute of Mathematics, Hanoi, Vietnam.     

Hidden convexity in some nonconvex quadratically constrained quadratic programming

We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints. Under a suitable simultaneous diagonalization assumption (which trivially holds for trust region type problems), we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints, which is also shown to be equivalent to a minimax convex problem. In both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts. Special cases and applications are also discussed. We outline interior-point polynomial-time algorithms for the solution of the equivalent convex programs. This author's work was partially supported by GIF, the German-Israeli Foundation for Scientific Research and Development and by the Binational Science Foundation. This author's work was partially supported by National Science Foundation Grants DMS-9201297 and DMS-9401871.     

Global solution of optimization problems with signomial parts

In this paper a new approach for the global solution of nonconvex MINLP (Mixed Integer NonLinear Programming) problems that contain signomial (generalized geometric) expressions is proposed and illustrated. By applying different variable transformation techniques and a discretization scheme a lower bounding convex MINLP problem can be derived. The convexified MINLP problem can be solved with standard methods. The key element in this approach is that all transformations are applied termwise. In this way all convex parts of the problem are left unaffected by the transformations. The method is illustrated by four example problems.     

Convergence and Application of a Decomposition Method Using Duality Bounds for Nonconvex Global Optimization

The subject of this article is a class of global optimization problems, in which the variables can be divided into two groups such that, in each group, the functions involved have the same structure (e.g. linear, convex or concave, etc.). Based on the decomposition idea of Benders (Ref. 1), a corresponding master problem is defined on the space of one of the two groups of variables. The objective function of this master problem is in fact the optimal value function of a nonlinear parametric optimization problem. To solve the resulting master problem, a branch-and-bound scheme is proposed, in which the estimation of the lower bounds is performed by applying the well-known weak duality theorem in Lagrange duality. The results of this article concentrate on two subjects: investigating the convergence of the general algorithm and solving dual problems of some special classes of nonconvex optimization problems. Based on results in sensitivity and stability theory and in parametric optimization, conditions for the convergence are established by investigating the so-called dual properness property and the upper semicontinuity of the objective function of the master problem. The general algorithm is then discussed in detail for some nonconvex problems including concave minimization problems with a special structure, general quadratic problems, optimization problems on the efficient set, and linear multiplicative programming problems.     

Convexification procedures and decomposition methods for nonconvex optimization problems

In order for primal-dual methods to be applicable to a constrained minimization problem, it is necessary that restrictive convexity conditions are satisfied. In this paper, we consider a procedure by means of which a nonconvex problem is convexified and transformed into one which can be solved with the aid of primal-dual methods. Under this transformation, separability of the type necessary for application of decomposition algorithms is preserved. This feature extends the range of applicability of such algorithms to nonconvex problems. Relations with multiplier methods are explored with the aid of a local version of the notion of a conjugate convex function.This work was carried out at the Coordinated Science Laboratory, University of Illinois, Urbana, Illinois, and was supported by the National Science Foundation under Grant ENG 74-19332.     

The complementary convex structure in global optimization

We show the importance of exploiting the complementary convex structure for efficiently solving a wide class of specially structured nonconvex global optimization problems. Roughly speaking, a specific feature of these problems is that their nonconvex nucleus can be transformed into a complementary convex structure which can then be shifted to a subspace of much lower dimension than the original underlying space. This approach leads to quite efficient algorithms for many problems of practical interest, including linear and convex multiplicative programming problems, concave minimization problems with few nonlinear variables, bilevel linear optimization problems, etc...     

Beyond Convex? Global Optimization is Feasible Only for Convex Objective Functions: A Theorem

It is known that there are feasible algorithms for minimizing convex functions, and that for general functions, global minimization is a difficult (NP-hard) problem. It is reasonable to ask whether there exists a class of functions that is larger than the class of all convex functions for which we can still solve the corresponding minimization problems feasibly. In this paper, we prove, in essence, that no such more general class exists. In other words, we prove that global optimization is always feasible only for convex objective functions.     

An outer approximation method for minimizing the product of several convex functions on a convex set

This paper addresses the minimization of the product ofp convex functions on a convex set. It is shown that this nonconvex problem can be converted to a concave minimization problem withp variables, whose objective function value is determined by solving a convex minimization problem. An outer approximation method is proposed for obtaining a global minimum of the resulting problem. Computational experiments indicate that this algorithm is reasonable efficient whenp is less than 4.This research was partly supported by Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture, Grant No. (C)03832018 and (C)04832010.     

Canonical Dual Transformation Method and Generalized Triality Theory in Nonsmooth Global Optimization*

This paper presents, within a unified framework, a potentially powerful canonical dual transformation method and associated generalized duality theory in nonsmooth global optimization. It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in ⁿ can be reformulated into certain smooth/convex unconstrained dual problems in ^m with m n and without duality gap, and some NP-hard concave minimization problems can be transformed into unconstrained convex minimization dual problems. The extended Lagrange duality principles proposed recently in finite deformation theory are generalized suitable for solving a large class of nonconvex and nonsmooth problems. The very interesting generalized triality theory can be used to establish nice theoretical results and to develop efficient alternative algorithms for robust computations.     

Nonlinear rescaling and proximal-like methods in convex optimization

The nonlinear rescaling principle (NRP) consists of transforming the objective function and/or the constraints of a given constrained optimization problem into another problem which is equivalent to the original one in the sense that their optimal set of solutions coincides. A nonlinear transformation parameterized by a positive scalar parameter and based on a smooth sealing function is used to transform the constraints. The methods based on NRP consist of sequential unconstrained minimization of the classical Lagrangian for the equivalent problem, followed by an explicit formula updating the Lagrange multipliers. We first show that the NRP leads naturally to proximal methods with an entropy-like kernel, which is defined by the conjugate of the scaling function, and establish that the two methods are dually equivalent for convex constrained minimization problems. We then study the convergence properties of the nonlinear rescaling algorithm and the corresponding entropy-like proximal methods for convex constrained optimization problems. Special cases of the nonlinear rescaling algorithm are presented. In particular a new class of exponential penalty-modified barrier functions methods is introduced. Partially supported by the National Science Foundation, under Grants DMS-9201297, and DMS-9401871. Partially supported by NASA Grant NAG3-1397 and NSF Grant DMS-9403218.     

Generalized linear multiplicative and fractional programming

This paper addresses itself to the algorithm for minimizing the sum of a convex function and a product of two linear functions over a polytope. It is shown that this nonconvex minimization problem can be solved by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in higher dimension and apply a parametric programming (path following) approach. Also it is shown that the same idea can be applied to a generalized linear fractional programming problem whose objective function is the sum of a convex function and a linear fractional function.     

Solutions to quadratic minimization problems with box and integer constraints

This paper presents a canonical duality theory for solving quadratic minimization problems subjected to either box or integer constraints. Results show that under Gao and Strang’s general global optimality condition, these well-known nonconvex and discrete problems can be converted into smooth concave maximization dual problems over closed convex feasible spaces without duality gap, and can be solved by well-developed optimization methods. Both existence and uniqueness of these canonical dual solutions are presented. Based on a second-order canonical dual perturbation, the discrete integer programming problem is equivalent to a continuous unconstrained Lipschitzian optimization problem, which can be solved by certain deterministic technique. Particularly, an analytical solution is obtained under certain condition. A fourth-order canonical dual perturbation algorithm is presented and applications are illustrated. Finally, implication of the canonical duality theory for the popular semi-definite programming method is revealed.     

Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization

It is shown that, for very general classes of nonconvex global optimization problems, the duality gap obtained by solving a corresponding Lagrangian dual in reduced to zero in the limit when combined with suitably refined partitioning of the feasible set. A similar result holds for partly convex problems where exhaustive partitioning is applied only in the space of nonconvex variables. Applications include branch-and-bound approaches for linearly constrained problems where convex envelopes can be computed, certain generalized bilinear problems, linearly constrained optimization of the sum of ratios of affine functions, and concave minimization under reverse convex constraints.     

Local Saddle Points and Convexification for Nonconvex Optimization Problems

Recently, Li (Ref. 1) obtained a saddle-point result for a general class of nonconvex optimization problems with inequality constraints, by using a transformation equivalent to taking the pth power of the objective function and the constraints, under several conditions. In this paper, we show that, by an equivalent transformation, using the pth power of the constraints or using the pth power of the objective function and the constraints, the same result can be obtained under much weaker and reasonable conditions. Also, our results can be extended to the problem in which equality and inequality constraints are involved.     

A class of convexification and concavification methods for non-monotone optimization problems

A class of convexification and concavification methods are proposed for solving some classes of non-monotone optimization problems. It is shown that some classes of non-monotone optimization problems can be converted into better structured optimization problems, such as, concave minimization problems, reverse convex programming problems, and canonical D.C. programming problems by the proposed convexification and concavification methods. The equivalence between the original problem and the converted better structured optimization problem is established.     

On the convergence of asynchronous parallel algorithm for large-scale linearly constrained minimization problem

As a synchronization parallel framework, the parallel variable transformation (PVT) algorithm is effective to solve unconstrained optimization problems. In this paper, based on the idea that a constrained optimization problem is equivalent to a differentiable unconstrained optimization problem by introducing the Fischer Function, we propose an asynchronous PVT algorithm for solving large-scale linearly constrained convex minimization problems. This new algorithm can terminate when some processor satisfies terminal condition without waiting for other processors. Meanwhile, it can enhances practical efficiency for large-scale optimization problem. Global convergence of the new algorithm is established under suitable assumptions. And in particular, the linear rate of convergence does not depend on the number of processors.     

单调全局最优化问题的凸化外逼近算法

单调优化是指目标函数与约束函数均为单调函数的全局优化问题．本文提出一种新的凸化变换方法把单调函数化为凸函数，进而把单调优化问题化为等价的凸极大或凹极小问题，然后采用Hoffman的外逼近方法来求得问题的全局最优解．我们把这种凸化方法同Tuy的Polyblock外逼近方法作了比较，通过数值比较可以看出本文提出的凸化的方法在收敛速度上明显优于Polyblock方法．