An interior Newton method for quadratic programming

We propose a new (interior) approach for the general quadratic programming problem. We establish that the new method has strong convergence properties: the generated sequence converges globally to a point satisfying the second-order necessary optimality conditions, and the rate of convergence is 2-step quadratic if the limit point is a strong local minimizer. Published alternative interior approaches do not share such strong convergence properties for the nonconvex case. We also report on the results of preliminary numerical experiments: the results indicate that the proposed method has considerable practical potential. Received October 11, 1993 / Revised version received February 20, 1996 Published online July 19, 1999     

A trust region and affine scaling interior point method for nonconvex minimization with linear inequality constraints

A trust region and affine scaling interior point method (TRAM) is proposed for a general nonlinear minimization with linear inequality constraints [8]. In the proposed approach, a Newton step is derived from the complementarity conditions. Based on this Newton step, a trust region subproblem is formed, and the original objective function is monotonically decreased. Explicit sufficient decrease conditions are proposed for satisfying the first order and second order necessary conditions.?The objective of this paper is to establish global and local convergence properties of the proposed trust region and affine scaling interior point method. It is shown that the proposed explicit decrease conditions are sufficient for satisfy complementarity, dual feasibility and second order necessary conditions respectively. It is also established that a trust region solution is asymptotically in the interior of the proposed trust region subproblem and a properly damped trust region step can achieve quadratic convergence. Received: January 29, 1999 / Accepted: November 22, 1999?Published online February 23, 2000     

Wedge trust region methods for derivative free optimization

A new method for derivative-free optimization is presented. It is designed for solving problems in which the objective function is smooth and the number of variables is moderate, but the gradient is not available. The method generates a model that interpolates the objective function at a set of sample points, and uses trust regions to promote convergence. The step-generation subproblem ensures that all the iterates satisfy a geometric condition and are therefore adequate for updating the model. The sample points are updated using a scheme that improves the accuracy of the interpolation model when needed. Two versions of the method are presented: one using linear models and the other using quadratic models. Numerical tests comparing the new approach with established methods for derivate-free optimization are reported. Received: October 2000 / Accepted: August 2001?Published online October 26, 2001     

Failure of global convergence for a class of interior point methods for nonlinear programming

Using a simple analytical example, we demonstrate that a class of interior point methods for general nonlinear programming, including some current methods, is not globally convergent. It is shown that those algorithms produce limit points that are neither feasible nor stationary points of some measure of the constraint violation, when applied to a well-posed problem. Received: December 1999 / Accepted: May 2000?Published online August 18, 2000     

Properties of an interior embedding for solving nonlinear optimization problems

The paper presents an interior embedding of nonlinear optimization problems. This embedding satisfies a sufficient condition for the success of pathfollowing algorithms with jumps being applied to one-parametric optimization problems.?The one-parametric problem obtained by the embedding is supposed to be regular in the sense of Jongen, Jonker and Twilt. This asumption is analyzed, and its genericity is proved in the space of the original optimization problems. Received May 20, 1997 / Revised version received October 6, 1998?Published online May 12, 1999     

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     

On maximization of quadratic form over intersection of ellipsoids with common center

We demonstrate that if A ₁,...,A _m are symmetric positive semidefinite n×n matrices with positive definite sum and A is an arbitrary symmetric n×n matrix, then the relative accuracy, in terms of the optimal value, of the semidefinite relaxation of the optimization program is not worse than . It is shown that this bound is sharp in order, as far as the dependence on m is concerned, and that a~feasible solution x to (P) with can be found efficiently. This somehow improves one of the results of Nesterov [4] where bound similar to (*) is established for the case when all A_i are of rank 1. Received August 13, 1998 / Revised version received May 25, 1999? Published online September 15, 1999     

A primal interior point method for the linear semidefinite programming problem

The linear semidefinite programming problem is examined. A primal interior point method is proposed to solve this problem. It extends the barrier-projection method used for linear programs. The basic properties of the proposed method are discussed, and its local convergence is proved.     

An infeasible-interior-point potential-reduction algorithm for linear programming

n . The method is based on Rockafellar’s proximal point algorithm and a cutting-plane technique. At each step, we use an approximate proximal point p^a(x_k) of x_k to define a v_k∈∂_εkf(p^a(x_k)) with ε_k≤α∥v_k∥, where α is a constant. The method monitors the reduction in the value of ∥v_k∥ to identify when a line search on f should be used. The quasi-Newton step is used to reduce the value of ∥v_k∥. Without the differentiability of f, the method converges globally and the rate of convergence is Q-linear. Superlinear convergence is also discussed to extend the characterization result of Dennis and Moré. Numerical results show the good performance of the method. Received October 3, 1995 / Revised version received August 20, 1998 Published online January 20, 1999     

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