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 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 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     

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.     

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 stable interior penalty method for convex extremal problems

Summary For a nonlinearly constrained convex extremal problem a general interior penalty method is given, that is Hadamard-stable and needs no compactness conditions for convergence. The rate of convergence of the values iso(t) fort+0.     

Partly convex programming and zermelo's navigation problems

Mathematical programs, that become convex programs after freezing some variables, are termed partly convex. For such programs we give saddle-point conditions that are both necessary and sufficient that a feasible point be globally optimal. The conditions require cooperation of the feasible point tested for optimality, an assumption implied by lower semicontinuity of the feasible set mapping. The characterizations are simplified if certain point-to-set mappings satisfy a sandwich condition.The tools of parametric optimization and basic point-to-set topology are used in formulating both optimality conditions and numerical methods. In particular, we solve a large class of Zermelo's navigation problems and establish global optimality of the numerical solutions.Research partly supported by NSERC of Canada.     

Robust duality for generalized convex programming problems under data uncertainty

In this paper we present a robust duality theory for generalized convex programming problems in the face of data uncertainty within the framework of robust optimization. We establish robust strong duality for an uncertain nonlinear programming primal problem and its uncertain Lagrangian dual by showing strong duality between the deterministic counterparts: robust counterpart of the primal model and the optimistic counterpart of its dual problem. A robust strong duality theorem is given whenever the Lagrangian function is convex. We provide classes of uncertain non-convex programming problems for which robust strong duality holds under a constraint qualification. In particular, we show that robust strong duality is guaranteed for non-convex quadratic programming problems with a single quadratic constraint with the spectral norm uncertainty under a generalized Slater condition. Numerical examples are given to illustrate the nature of robust duality for uncertain nonlinear programming problems. We further show that robust duality continues to hold under a weakened convexity condition.     

A predictor-corrector smoothing Newton method for symmetric cone complementarity problems

In this paper, we present a predictor-corrector smoothing Newton method for solving nonlinear symmetric cone complementarity problems (SCCP) based on the symmetrically perturbed smoothing function. Under a mild assumption, the solution set of the problem concerned is just nonempty, we show that the proposed algorithm is globally and locally quadratic convergent. Also, the algorithm finds a maximally complementary solution to the SCCP. Numerical results for second order cone complementarity problems (SOCCP), a special case of SCCP, show that the proposed algorithm is effective.     

A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems

Summary In this paper, we shall be concerned with the solution of constrained convex minimization problems. The constrained convex minimization problems are proposed to be transformable into a convex-additively decomposed and almost separable form, e.g. by decomposition of the objective functional and the restrictions. Unconstrained dual problems are generated by using Fenchel-Rockafellar duality. This decomposition-dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable. Moreover, the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return-formulas. In quadratic programming the decomposition-dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated byH-matrices. Numerical tests for solving obstacle problems in ¹ discretized by using piecewise quadratic finite elements and in ² by using the five-point difference approximation are presented.     

A decomposition method for convex minimization problems and its application

1. IntroductionConsider the following special convex programming problem(P) adn{f(~) g(z); Ax = z},where f: Re - (--co, co] and g: Re - (--co, co] are closed proper convex functions andA is an m x n matrix. The Lagrangian for problem (P) is defined by L: Rad x Re x Re -- (~co, co] as follows:L(x, z, y) = f(x) g(z) (y, Ax ~ z), (1.1)where (., .) denotes the inner product in the general sense and 'y is the Lagrangian multiplierassociated with the constraint Ax = z. The augmented L…     

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 simple constraint qualification in convex programming

We introduce and characterize a class of differentiable convex functions for which the Karush—Kuhn—Tucker condition is necessary for optimality. If some constraints do not belong to this class, then the characterization of optimality generally assumes an asymptotic form.We also show that for the functions that belong to this class in multi-objective optimization, Pareto solutions coincide with strong Pareto solutions,. This extends a result, well known for the linear case.Research partly supported by the National Research Council of Canada.     

A quasi-linear algorithm for calculating the infimal convolution of convex quadratic functions

In this paper we present an algorithm of quasi-linear complexity to exactly calculate the infimal convolution of convex quadratic functions. The algorithm exactly and simultaneously solves a separable uniparametric family of quadratic programming problems resulting from varying the equality constraint.     

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 branch-and-cut method for 0-1 mixed convex programming

We generalize the disjunctive approach of Balas, Ceria, and Cornuéjols [2] and devevlop a branch-and-cut method for solving 0-1 convex programming problems. We show that cuts can be generated by solving a single convex program. We show how to construct regions similar to those of Sherali and Adams [20] and Lovász and Schrijver [12] for the convex case. Finally, we give some preliminary computational results for our method. Received January 16, 1996 / Revised version received April 23, 1999?Published online June 28, 1999     

A filled function method for global optimization

A novel filled function with one parameter is suggested in this paper for finding a global minimizer for a general class of nonlinear programming problems with a closed bounded box. A new algorithm is presented according to the theoretical analysis. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.     

On a generalized proximal point method for solving equilibrium problems in Banach spaces

We introduce a regularized equilibrium problem in Banach spaces, involving generalized Bregman functions. For this regularized problem, we establish the existence and uniqueness of solutions. These regularizations yield a proximal-like method for solving equilibrium problems in Banach spaces. We prove that the proximal sequence is an asymptotically solving sequence when the dual space is uniformly convex. Moreover, we prove that all weak accumulation points are solutions if the equilibrium function is lower semicontinuous in its first variable. We prove, under additional assumptions, that the proximal sequence converges weakly to a solution.     

A spatial domain decomposition method for parabolic optimal control problems

We present a non-overlapping spatial domain decomposition method for the solution of linear–quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space–time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space–time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space–time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann–Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou.