A proximal-based decomposition method for convex minimization problems

This paper presents a decomposition method for solving convex minimization problems. At each iteration, the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables. We derive this algorithm from Rockafellar's proximal method of multipliers, which involves an augmented Lagrangian with an additional quadratic proximal term. The algorithm preserves the good features of the proximal method of multipliers, with the additional advantage that it leads to a decoupling of the constraints, and is thus suitable for parallel implementation. We allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem's data, the method is globally convergent and at a linear rate. The method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets.Corresponding author. Partially supported by Air Force Office of Scientific Research Grant 91-0008 and National Science Foundation Grant DMS-9201297.     

Unconstrained minimization approaches to nonlinear complementarity problems

The nonlinear complementarity problem can be reformulated as unconstrained minimization problems by introducing merit functions. Under some assumptions, the solution set of the nonlinear complementarity problem coincides with the set of local minima of the corresponding minimization problem. These results were presented by Mangasarian and Solodov, Yamashita and Fukushima, and Geiger and Kanzow. In this note, we generalize some results of Mangasarian and Solodov, Yamashita and Fukushima, and Geiger and Kanzow to the case where the considered function is only directionally differentiable. Some results are strengthened in the smooth case. For example, it is shown that the strong monotonicity condition can be replaced by the P-uniform property for ensuring a stationary point of the reformulated unconstrained minimization problems to be a solution of the nonlinear complementarity problem. We also present a descent algorithm for solving the nonlinear complementarity problem in the smooth case. Any accumulation point generated by this algorithm is proved to be a solution of the nonlinear complementarity under the monotonicity condition.     

Convergence of some algorithms for convex minimization

We present a simple and unified technique to establish convergence of various minimization methods. These contain the (conceptual) proximal point method, as well as implementable forms such as bundle algorithms, including the classical subgradient relaxation algorithm with divergent series.An important research work of Phil Wolfe's concerned convex minimization. This paper is dedicated to him, on the occasion of his 65th birthday, in appreciation of his creative and pioneering work.     

A proximal bundle method with inexact data for convex nondifferentiable minimization

A proximal bundle method with inexact data is presented for minimizing an unconstrained nonsmooth convex function f$f$. At each iteration, only the approximate evaluations of f$f$ and its ε$\epsilon$-subgradients are required and its search directions are determined via solving quadratic programmings. Compared with the pre-existing results, the polyhedral approximation model that we offer is more precise and a new term is added into the estimation term of the descent from the model. It is shown that every cluster of the sequence of iterates generated by the proposed algorithm is an exact solution of the unconstrained minimization problem.     

A constraint linearization method for nondifferentiable convex minimization

Summary This paper presents a readily implementable algorithm for solving constrained minimization problems involving (possibly nonsmooth) convex functions. The constraints are handled as in the successive quadratic approximations methods for smooth problems. An exact penalty function is employed for stepsize selection. A scheme for automatic limitation of penalty growth is given. Global convergence of the algorithm is established, as well as finite termination for piecewise linear problems. Numerical experience is reported.Sponsored by Program CPBP 02.15     

A parametric approach for solving a class of generalized quadratic-transformable rank-two nonconvex programs

The aim of this paper is to propose a solution algorithm for a particular class of rank-two nonconvex programs having a polyhedral feasible region. The algorithm is based on the so-called “optimal level solutions” method. Various global optimality conditions are discussed and implemented in order to improve the efficiency of the algorithm.     

Global convergence of the nonmonotone MBFGS method for nonconvex unconstrained minimization

In this paper, we propose a new nonmonotone Armijo type line search and prove that the MBFGS method proposed by Li and Fukushima with this new line search converges globally for nonconvex minimization. Some numerical experiments show that this nonmonotone MBFGS method is efficient for the given test problems.     

Convex separable minimization with box constraints

A minimization problem with convex separable objective function subject to a convex separable inequality constraint of the form “less than or equal to” and bounds on the variables (box constraints) is considered. Necessary and sufficient condition is proved for a feasible solution to be an optimal solution to this problem. An iterative algorithm of polynomial complexity for solving problems of the considered form is suggested and its convergence is proved. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization

Bisubmodular functions are a natural “directed”, or “signed”, extension of submodular functions with several applications. Recently Fujishige and Iwata showed how to extend the Iwata, Fleischer, and Fujishige (IFF) algorithm for submodular function minimization (SFM) to bisubmodular function minimization (BSFM). However, they were able to extend only the weakly polynomial version of IFF to BSFM. Here we investigate the difficulty that prevented them from also extending the strongly polynomial version of IFF to BSFM, and we show a way around the difficulty. This new method gives a somewhat simpler strongly polynomial SFM algorithm, as well as the first combinatorial strongly polynomial algorithm for BSFM. This further leads to extending Iwata’s fully combinatorial version of IFF to BSFM. The research of S. T. McCormick was supported by an NSERC Operating Grant. The research of S. Fujishige was supported by a Grant-in-Aid of the Ministry of Education, Culture, Science and Technology of Japan.     

A variable metric algorithm for unconstrained minimization without evaluation of derivatives

Summary This paper presents a modification of the BFGS-method for unconstrained minimization that avoids computation of derivatives. The gradients are approximated by the aid of differences of function values. These approximations are calculated in such a way that a complete convergence proof can be given. The presented algorithm is implementable, no exact line search is required. It is shown that, if the objective function is convex and some usually required conditions hold, the algorithm converges to a solution. If the Hessian matrix of the objective function is positive definite and satisfies a Lipschitz-condition in a neighbourhood of the solution, then the rate of convergence is superlinear.     

A row-action method for convex programming

We present a new method for minimizing a strictly convex function subject to general convex constraints. Constraints are used one at a time, no changes are made in the constraint functions (thus the row-action nature of the algorithm) and at each iteration a subproblem is solved consisting of minimization of the objective function subject to one or two linear equations. Convergence of the algorithm is established and the method is compared with other row-action algorithms for several relevant particular cases.Corresponding author. Research of this author was partially supported by CNPq grant No. 301280/86.     

Necessary optimality conditions for bilevel minimization problems

In this paper we study bilevel minimization problems. Using the implicit function theorem, variational analysis and exact penalty results we establish necessary optimality conditions for these problems.     

Reformulation in mathematical programming: An application to quantum chemistry

This paper concerns the application of reformulation techniques in mathematical programming to a specific problem arising in quantum chemistry, namely the solution of Hartree-Fock systems of equations, which describe atomic and molecular electronic wave functions based on the minimization of a functional of the energy. Their traditional solution method does not provide a guarantee of global optimality and its output depends on a provided initial starting point. We formulate this problem as a multi-extremal nonconvex polynomial programming problem, and solve it with a spatial Branch-and-Bound algorithm for global optimization. The lower bounds at each node are provided by reformulating the problem in such a way that its convex relaxation is tight. The validity of the proposed approach was established by successfully computing the ground-state of the helium and beryllium atoms.     

Affine scaling interior Levenberg–Marquardt method for bound-constrained semismooth equations under local error bound conditions

We develop and analyze a new affine scaling Levenberg–Marquardt method with nonmonotonic interior backtracking line search technique for solving bound-constrained semismooth equations under local error bound conditions. The affine scaling Levenberg–Marquardt equation is based on a minimization of the squared Euclidean norm of linear model adding a quadratic affine scaling matrix to find a solution that belongs to the bounded constraints on variable. The global convergence results are developed in a very general setting of computing trial directions by a semismooth Levenberg–Marquardt method where a backtracking line search technique projects trial steps onto the feasible interior set. We establish that close to the solution set the affine scaling interior Levenberg–Marquardt algorithm is shown to converge locally Q-superlinearly depending on the quality of the semismooth and Levenberg–Marquardt parameter under an error bound assumption that is much weaker than the standard nonsingularity condition, that is, BD-regular condition under nonsmooth case. A nonmonotonic criterion should bring about speed up the convergence progress in the contours of objective function with large curvature.     

Descent methods for quasidifferentiable minimization

A descent method is given for minimizing a nondifferentiable function which can be locally approximated by pointwise minima of convex functions. At each iterate the algorithm finds several directions by solving several linear or quadratic programming subproblems. These directions are then used in an Armijo-like search for the next iterate. A feasible direction extension to inequality constrained minimization problems is also presented. The algorithms converge to points satisfying necessary optimality conditions which are sharper than the ones involved in convergence results for algorithms based on the Clarke subdifferential.This research was sponsored by Project 02.15.     

A method of linearizations for linearly constrained nonconvex nonsmooth minimization

A readily implementable algorithm is given for minimizing a (possibly nondifferentiable and nonconvex) locally Lipschitz continuous functionf subject to linear constraints. At each iteration a polyhedral approximation tof is constructed from a few previously computed subgradients and an aggregate subgradient, which accumulates the past subgradient information. This aproximation and the linear constraints generate constraints in the search direction finding subproblem that is a quadratic programming problem. Then a stepsize is found by an approximate line search. All the algorithm's accumulation points are stationary. Moreover, the algorithm converges whenf happens to be convex.     

A cut-peak function method for global optimization

A new method is proposed for solving box constrained global optimization problems. The basic idea of the method is described as follows: Constructing a so-called cut-peak function and a choice function for each present minimizer, the original problem of finding a global solution is converted into an auxiliary minimization problem of finding local minimizers of the choice function, whose objective function values are smaller than the previous ones. For a local minimum solution of auxiliary problems this procedure is repeated until no new minimizer with a smaller objective function value could be found for the last minimizer. Construction of auxiliary problems and choice of parameters are relatively simple, so the algorithm is relatively easy to implement, and the results of the numerical tests are satisfactory compared to other methods.     

A simplicial branch and duality bound algorithm for the sum of convex–convex ratios problem

This article presents a simplicial branch and duality bound algorithm for globally solving the sum of convex–convex ratios problem with nonconvex feasible region. To our knowledge, little progress has been made for globally solving this problem so far. The algorithm uses a branch and bound scheme where the Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems during the algorithm search are all ordinary linear programs that can be solved very efficiently. It has been proved that the algorithm possesses global convergence. Finally, the numerical experiments are given to show the feasibility of the proposed algorithm.     

Quasi-concave minimization subject to linear constraints

A finite algorithm is presented for solving the quasi-concave minimization problem subject to linear constraints. The concept of an extreme point is generalized to that of an extreme facet of a polyhedron. Then a search routine is developed for the detection of an extreme facet of the feasible region relative to the polyhedron defined by the current set of cuts. After identifying an extreme facet we cut it off by a cut developed for this purpose. We call this cut the facet cut. The method is both compatible with other cutting procedures and is finite..