A Global Optimization Method,QBB, for Twice-Differentiable Nonconvex Optimization Problem

A global optimization method, QBB, for twice-differentiable NLPs (Non-Linear Programming) is developed to operate within a branch-and-bound framework and require the construction of a relaxed convex problem on the basis of the quadratic lower bounding functions for the generic nonconvex structures. Within an exhaustive simplicial division of the constrained region, the rigorous quadratic underestimation function is constructed for the generic nonconvex function structure by virtue of the maximal eigenvalue analysis of the interval Hessian matrix. Each valid lower bound of the NLP problem with the division progress is computed by the convex programming of the relaxed optimization problem obtained by preserving the convex or linear terms, replacing the concave term with linear convex envelope, underestimating the special terms and the generic terms by using their customized tight convex lower bounding functions or the valid quadratic lower bounding functions, respectively. The standard convergence properties of the QBB algorithm for nonconvex global optimization problems are guaranteed. The preliminary computation studies are presented in order to evaluate the algorithmic efficiency of the proposed QBB approach.     

Duality Principles for Optimization Problems Dealing with the Difference of Vector-Valued Convex Mappings

Using the concept of a subdifferential of a vector-valued convex mapping, we provide duality formulas for the minimization of nonconvex composite functions and related optimization problems such as the minimization of a convex function over a vectorial DC constraint.     

Bundle-based descent method for nonsmooth multiobjective DC optimization with inequality constraints

Multiobjective DC optimization problems arise naturally, for example, in data classification and cluster analysis playing a crucial role in data mining. In this paper, we propose a new multiobjective double bundle method designed for nonsmooth multiobjective optimization problems having objective and constraint functions which can be presented as a difference of two convex (DC) functions. The method is of the descent type and it generalizes the ideas of the double bundle method for multiobjective and constrained problems. We utilize the special cutting plane model angled for the DC improvement function such that the convex and the concave behaviour of the function is captured. The method is proved to be finitely convergent to a weakly Pareto stationary point under mild assumptions. Finally, we consider some numerical experiments and compare the solutions produced by our method with the method designed for general nonconvex multiobjective problems. This is done in order to validate the usage of the method aimed specially for DC objectives instead of a general nonconvex method.     

Computing proximal points of nonconvex functions

The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with the critical points of the original function. This suggests that the many uses of proximal points, and their corresponding proximal envelopes (Moreau envelopes), will have a natural extension from convex optimization to nonconvex optimization. For example, the inexact proximal point methods for convex optimization might be redesigned to work for nonconvex functions. In order to begin the practical implementation of proximal points in a nonconvex setting, a first crucial step would be to design efficient methods of approximating nonconvex proximal points. This would provide a solid foundation on which future design and analysis for nonconvex proximal point methods could flourish. In this paper we present a methodology based on the computation of proximal points of piecewise affine models of the nonconvex function. These models can be built with only the knowledge obtained from a black box providing, for each point, the function value and one subgradient. Convergence of the method is proved for the class of nonconvex functions that are prox-bounded and lower- ${\mathcal{C}}^2$ and encouraging preliminary numerical testing is reported.     

A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called αBB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.     

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.     

Hidden Convex Minimization

A class of nonconvex minimization problems can be classified as hidden convex minimization problems. A nonconvex minimization problem is called a hidden convex minimization problem if there exists an equivalent transformation such that the equivalent transformation of it is a convex minimization problem. Sufficient conditions that are independent of transformations are derived in this paper for identifying such a class of seemingly nonconvex minimization problems that are equivalent to convex minimization problems. Thus, a global optimality can be achieved for this class of hidden convex optimization problems by using local search methods. The results presented in this paper extend the reach of convex minimization by identifying its equivalent with a nonconvex representation.     

A sequential parametric convex approximation method with applications to nonconvex truss topology design problems

We describe a general scheme for solving nonconvex optimization problems, where in each iteration the nonconvex feasible set is approximated by an inner convex approximation. The latter is defined using an upper bound on the nonconvex constraint functions. Under appropriate conditions, a monotone convergence to a KKT point is established. The scheme is applied to truss topology design (TTD) problems, where the nonconvex constraints are associated with bounds on displacements and stresses. It is shown that the approximate convex problem solved at each inner iteration can be cast as a conic quadratic programming problem, hence large scale TTD problems can be efficiently solved by the proposed method.     

Solving an Inverse Problem for an Elliptic Equation by d.c. Programming

An inverse problem of determination of a coefficient in an elliptic equation is considered. This problem is ill-posed in the sense of Hadamard and Tikhonov's regularization method is used for solving it in a stable way. This method requires globally solving nonconvex optimization problems, the solution methods for which have been very little studied in the inverse problems community. It is proved that the objective function of the corresponding optimization problem for our inverse problem can be represented as the difference of two convex functions (d.c. functions), and the difference of convex functions algorithm (DCA) in combination with a branch-and-bound technique can be used to globally solve it. Numerical examples are presented which show the efficiency of the method.     

Limit vector variational inequality problems via scalarization

This paper presents a global optimization approach for solving signomial geometric programming (SGP) problems. We employ an accelerated extended cutting plane (ECP) approach integrated with piecewise linear (PWL) approximations to solve the global optimization of SGP problems. In this approach, we separate the feasible regions determined by the constraints into convex and nonconvex ones in the logarithmic domain. In the nonconvex feasible regions, the corresponding constraint functions are converted into mixed integer linear constraints using PWL approximations, while the other constraints with convex feasible regions are handled by the ECP method. We also use pre-processed initial cuts and batched cuts to accelerate the proposed algorithm. Numerical results show that the proposed approach can solve the global optimization of SGP problems efficiently and effectively.     

Nonconvex Piecewise-Quadratic Underestimation for Global Minimization

Motivated by the fact that important real-life problems, such as the protein docking problem, can be accurately modeled by minimizing a nonconvex piecewise-quadratic function, a nonconvex underestimator is constructed as the minimum of a finite number of strictly convex quadratic functions. The nonconvex underestimator is generated by minimizing a linear function on a reverse convex region and utilizes sample points from a given complex function to be minimized. The global solution of the piecewise-quadratic underestimator is known exactly and gives an approximation to the global minimum of the original function. Successive shrinking of the initial search region to which this procedure is applied leads to fairly accurate estimates, within 0.0060%, of the global minima of synthetic nonconvex functions for which the global minima are known. Furthermore, this process can approximate a nonconvex protein docking function global minimum within four-figure relative accuracy in six refinement steps. This is less than half the number of refinement steps required by previous models such as the convex kernel underestimator (Mangasarian et al., Computational Optimization and Applications, to appear) and produces higher accuracy here.     

Attractive force search algorithm for piecewise convex maximization problems

We consider mathematical programming problems with the so-called piecewise convex objective functions. A solution method for this interesting and important class of nonconvex problems is presented. This method is based on Newton??s law of universal gravitation, multicriteria optimization and Helly??s theorem on convex bodies. Numerical experiments using well known classes of test problems on piecewise convex maximization, convex maximization as well as the maximum clique problem show the efficiency of the approach.     

Peaceman–Rachford splitting for a class of nonconvex optimization problems

We study the applicability of the Peaceman–Rachford (PR) splitting method for solving nonconvex optimization problems. When applied to minimizing the sum of a strongly convex Lipschitz differentiable function and a proper closed function, we show that if the strongly convex function has a large enough strong convexity modulus and the step-size parameter is chosen below a threshold that is computable, then any cluster point of the sequence generated, if exists, will give a stationary point of the optimization problem. We also give sufficient conditions guaranteeing boundedness of the sequence generated. We then discuss one way to split the objective so that the proposed method can be suitably applied to solving optimization problems with a coercive objective that is the sum of a (not necessarily strongly) convex Lipschitz differentiable function and a proper closed function; this setting covers a large class of nonconvex feasibility problems and constrained least squares problems. Finally, we illustrate the proposed algorithm numerically.     

Computational Experience with a New Class of Convex Underestimators: Box-constrained NLP Problems

In Akrotirianakis and Floudas (2004) we presented the theoretical foundations of a new class of convex underestimators for C ² nonconvex functions. In this paper, we present computational experience with those underestimators incorporated within a Branch-and-Bound algorithm for box-conatrained problems. The algorithm can be used to solve global optimization problems that involve C ² functions. We discuss several ways of incorporating the convex underestimators within a Branch-and-Bound framework. The resulting Branch-and-Bound algorithm is then used to solve a number of difficult box-constrained global optimization problems. A hybrid algorithm is also introduced, which incorporates a stochastic algorithm, the Random-Linkage method, for the solution of the nonconvex underestimating subproblems, arising within a Branch-and-Bound framework. The resulting algorithm also solves efficiently the same set of test problems.     

Numerical solution of a linear bilevel problem

The linear bilevel programming problem in the optimistic formulation is studied. It is reduced to an optimization problem with a nonconvex constraint in the form of a d.c. function (that is, the difference of two convex functions). For this problem, local and global search methods are developed. Numerical experiments performed for numerous specially generated problems, including large-scale ones, demonstrate the efficiency of the proposed approach.     

DC approach to weakly convex optimization and nonconvex quadratic optimization problems

Motivated by weakly convex optimization and quadratic optimization problems, we first show that there is no duality gap between a difference of convex (DC) program over DC constraints and its associated dual problem. We then provide certificates of global optimality for a class of nonconvex optimization problems. As an application, we derive characterizations of robust solutions for uncertain general nonconvex quadratic optimization problems over nonconvex quadratic constraints.     

Quadratic Majorization for Nonconvex Loss with Applications to the Boosting Algorithm

Classical robust statistical methods dealing with noisy data are often based on modifications of convex loss functions. In recent years, nonconvex loss-based robust methods have been increasingly popular. A nonconvex loss can provide robust estimation for data contaminated with outliers. The significant challenge is that a nonconvex loss can be numerically difficult to optimize. This article proposes quadratic majorization algorithm for nonconvex (QManc) loss. The QManc can decompose a nonconvex loss into a sequence of simpler optimization problems. Subsequently, the QManc is applied to a powerful machine learning algorithm: quadratic majorization boosting algorithm (QMBA). We develop QMBA for robust classification (binary and multi-category) and regression. In high-dimensional cancer genetics data and simulations, the QMBA is comparable with convex loss-based boosting algorithms for clean data, and outperforms the latter for data contaminated with outliers. The QMBA is also superior to boosting when directly implemented to optimize nonconvex loss functions. Supplementary material for this article is available online.     

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.     

A generalized Lagrangian function and multiplier method

As is well known, a saddle point for the Lagrangian function, if it exists, provides a solution to a convex programming problem; then, the values of the optimal primal and dual objective functions are equal. However, these results are not valid for nonconvex problems.In this paper, several results are presented on the theory of the generalized Lagrangian function, extended from the classical Lagrangian and the generalized duality program. Theoretical results for convex problems also hold for nonconvex problems by extension of the Lagrangian function. The concept of supporting hypersurfaces is useful to add a geometric interpretation to computational algorithms. This provides a basis to develop a new algorithm.     

The Global Convergence of Self-Scaling BFGS Algorithm with Nonmonotone Line Search for Unconstrained Nonconvex Optimization Problems

The self-scaling quasi-Newton method solves an unconstrained optimization problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessian approximation matrices of the objective function. It has been proved in the literature that this method has the global and superlinear convergence when the objective function is convex （or even uniformly convex）. We propose to solve unconstrained nonconvex optimization problems by a self-scaling BFGS algorithm with nonmonotone linear search. Nonmonotone line search has been recognized in numerical practices as a competitive approach for solving large-scale nonlinear problems. We consider two different nonmonotone line search forms and study the global convergence of these nonmonotone self-scale BFGS algorithms. We prove that, under some weaker condition than that in the literature, both forms of the self-scaling BFGS algorithm are globally convergent for unconstrained nonconvex optimization problems.