A bundle Bregman proximal method for convex nondifferentiable minimization

k } by taking x^k to be an approximate minimizer of , where is a piecewise linear model of f constructed from accumulated subgradient linearizations of f, D_h is the D-function of a generalized Bregman function h and t_k>0. Convergence under implementable criteria is established by extending our recent framework of Bregman proximal minimization, which is of independent interest, e.g., for nonquadratic multiplier methods for constrained minimization. In particular, we provide new insights into the convergence properties of bundle methods based on h=?|·|². Received September 18, 1997 / Revised version received June 30, 1998 Published online November 24, 1998     

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     

Metric subregularity for proximal generalized equations in Hilbert spaces

In this paper, we introduce and consider the concept of the prox-regularity of a multifunction. We mainly study the metric subregularity of a generalized equation defined by a proximal closed multifunction between two Hilbert spaces. Using proximal analysis techniques, we provide sufficient and/or necessary conditions for such a generalized equation to have the metric subregularity in Hilbert spaces. We also establish the results of Robinson-Ursescu theorem type for prox-regular multifunctions.     

Nonlinear error bounds for lower semicontinuous functions on metric spaces

As a development of the theory of linear error bounds for lower semicontinuous functions defined on complete metric spaces, introduced in Azé et al. (Nonlinear Anal 49, 643–670, 2002) and refined in Azé and Corvellec (ESAIM Control Optim Calc Var 10, 409–425, 2004), we propose a similar approach to nonlinear error bounds, based on the notion of strong slope, the variational principle, and the change-of-metric principle, the latter allowing to obtain sharp estimates for such error bounds through a reduction to the linear case.     

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.     

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.     

A cyclic iterative method for solving Multiple Sets Split Feasibility Problems in Banach Spaces

In this paper, we construct an iterative scheme and prove strong convergence theorem of the sequence generated to an approximate solution to a multiple sets split feasibility problem in a p-uniformly convex and uniformly smooth real Banach space. Some numerical experiments are given to study the efficiency and implementation of our iteration method. Our result complements the results of F. Wang (A new algorithm for solving the multiple-sets split feasibility problem in Banach spaces, Numerical Functional Anal. Optim. 35 (2014), 99–110), F. Scho¨pfer et al. (An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Problems 24 (2008), 055008) and many important recent results in this direction.     

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.     

Tikhonov-type regularization method for efficient solutions in vector optimization

The paper is devoted to developing the Tikhonov-type regularization algorithm of finding efficient solutions to the vector optimization problem for a mapping between finite dimensional Hilbert spaces with respect to the partial order induced by a pointed closed convex cone. We prove that under some suitable conditions either the sequence generated by our method converges to an efficient solution or all of its cluster points belong to the set of all efficient solutions of this problem.     

A method combining norm-relaxed QP subproblems with systems of linear equations for constrained optimization

Based on the ideas of norm-relaxed sequential quadratic programming (SQP) method and the strongly sub-feasible direction method, we propose a new SQP algorithm for the solution of nonlinear inequality constrained optimization. Unlike the previous work, at each iteration, the norm-relaxed quadratic programming subproblem (NRQPS) in our algorithm only consists of the constraints corresponding to an estimate of the active set, and the high-order correction direction (used to avoid the Maratos effect) is obtained by solving a system of linear equations (SLE) which also only consists of such a subset of constraints and gradients. Moreover, the line search technique can effectively combine the initialization process with the optimization process, and therefore (if the starting point is not feasible) the iteration points always get into the feasible set after a finite number of iterations. The global convergence is proved under the Mangasarian–Fromovitz constraint qualification (MFCQ), and the superlinear convergence is obtained without assuming the strict complementarity. Finally, the numerical experiments show that the proposed algorithm is effective and promising for the test problems.     

Optimality conditions for non-finite valued convex composite functions

Burke (1987) has recently developed second-order necessary and sufficient conditions for convex composite optimization in the case where the convex function is finite valued. In this note we present a technique for reducing the infinite valued case to the finite valued one. We then use this technique to extend the results in Burke (1987) to the case in which the convex function may take infinite values. We conclude by comparing these results with those established by Rockafellar (1989) for the piecewise linear-quadratic case.Dedicated to the memory of Robin W. ChaneyResearch supported in part by the National Science Foundation under grants DMS-8602399 and DMS-8803206, and by the Air Force Office of Scientific Research under grant ISSA-860080.Research supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983.     

A new trust region method for unconstrained optimization

In this paper, we propose a new trust region method for unconstrained optimization problems. The new trust region method can automatically adjust the trust region radius of related subproblems at each iteration and has strong global convergence under some mild conditions. We also analyze the global linear convergence, local superlinear and quadratic convergence rate of the new method. Numerical results show that the new trust region method is available and efficient in practical computation.     

A Model for Evaluation of Transport Policies in Multimodal Networks with Road and Parking Capacity Constraints

This paper presents a model for evaluation of transport policies in multimodal networks with road and parking capacity constraints. The proposed model simultaneously considers choices of travelers on route, parking location and mode between auto and transit. In the proposed model, it is assumed that auto drivers make a simultaneous route and parking location choice in a user equilibrium manner, and the modal split between auto and transit follows a multinomial logit formulation. A mathematical programming model with capacity constraints on road link and parking facilities is proposed that generates optimality conditions equivalent to the requirements for multimodal network equilibrium. An augmented Lagrangian dual algorithm embedded by partial linearization approach is developed to solve the proposed model. Numerical results on two example networks are presented to illustrate the proposed methodology. The results show that the service level of transit, parking charges, road link and parking capacities, and addition of a new parking location may bring significant impacts on travelers’ behavior and network performance. In addition, transport policies may result in paradoxical phenomenon.     

Weighted quasi-equilibrium problems with lower and upper bounds

This paper is devoted to studying the solution existence of weighted quasi-equilibrium problems with lower and upper bounds by using maximal element theorems, a fixed point theorem of set-valued mappings and Fan–KKM theorem, respectively. Some new results are obtained.     

A modified inexact implicit method for mixed variational inequalities

In this paper, we suggest and analyze an inexact implicit method with a variable parameter for mixed variational inequalities by using a new inexactness restriction. Under certain conditions, the global convergence of the proposed method is proved. Some preliminary computational results are given to illustrate the efficiency of the new inexactness restriction. The results proved in this paper may be viewed as improvement and refinement of the previously known results.     

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.     

An aggregate subgradient method for nonsmooth convex minimization

A class of implementable algorithms is described for minimizing any convex, not necessarily differentiable, functionf of several variables. The methods require only the calculation off and one subgradient off at designated points. They generalize Lemarechal's bundle method. More specifically, instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems, the methods use an aggregate subgradient which is recursively updated as the algorithms proceed. Each algorithm yields a minimizing sequence of points, and iff has any minimizers, then this sequence converges to a solution of the problem. Particular members of this algorithm class terminate whenf is piecewise linear. The methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user. Research sponsored by the Institute of Automatic Control, Technical University of Warsaw, Poland, under Project R.I.21.     

A self-adaptive trust region method with line search based on a simple subproblem model

In this paper, based on a simple model of the trust region subproblem, we propose a new self-adaptive trust region method with a line search technique for solving unconstrained optimization problems. By use of the simple subproblem model, the new method needs less memory capacitance and computational complexity. And the trust region radius is adjusted with a new self-adaptive adjustment strategy which makes full use of the information at the current point. When the trial step results in an increase in the objective function, the method does not resolve the subproblem, but it performs a line search technique from the failed point. Convergence properties of the method are proved under certain conditions. Numerical experiments show that the new method is effective and attractive for large-scale optimization problems.     

Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds

Local convergence analysis of the proximal point method for a special class of nonconvex functions on Hadamard manifold is presented in this paper. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, it is proved that each cluster point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minimizer is obtained.     

An augmented penalty function method with penalty parameter updates for nonconvex optimization

Given an augmented Lagrangian scheme for a general optimization problem, we use an epsilon subgradient step for improving the dual function. This can be seen as an update for an augmented penalty method, which is more stable because it does not force the penalty parameter to tend to infinity. We establish for this update primal-dual convergence for our augmented penalty method. As illustration, we apply our method to the test-bed kissing number problem.