Algorithms for bound constrained quadratic programming problems

Summary We present an algorithm which combines standard active set strategies with the gradient projection method for the solution of quadratic programming problems subject to bounds. We show, in particular, that if the quadratic is bounded below on the feasible set then termination occurs at a stationary point in a finite number of iterations. Moreover, if all stationary points are nondegenerate, termination occurs at a local minimizer. A numerical comparison of the algorithm based on the gradient projection algorithm with a standard active set strategy shows that on mildly degenerate problems the gradient projection algorithm requires considerable less iterations and time than the active set strategy. On nondegenerate problems the number of iterations typically decreases by at least a factor of 10. For strongly degenerate problems, the performance of the gradient projection algorithm deteriorates, but it still performs better than the active set method.Work supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38     

On the solution of concave knapsack problems

We consider a version of the knapsack problem which gives rise to a separable concave minimization problem subject to bounds on the variables and one equality constraint. We characterize strict local miniimizers of concave minimization problems subject to linear constraints, and use this characterization to show that although the problem of determining a global minimizer of the concave knapsack problem is NP-hard, it is possible to determine a local minimizer of this problem with at most O(n logn) operations and 1+[logn] evaluations of the function. If the function is quadratic this algorithm requires at most O(n logn) operations.Work supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38.Work supported in part by a Fannie and John Hertz Foundation graduate fellowship.     

Convergence properties of trust region methods for linear and convex constraints

We develop a convergence theory for convex and linearly constrained trust region methods which only requires that the step between iterates produce a sufficient reduction in the trust region subproblem. Global convergence is established for general convex constraints while the local analysis is for linearly constrained problems. The main local result establishes that if the sequence converges to a nondegenerate stationary point then the active constraints at the solution are identified in a finite number of iterations. As a consequence of the identification properties, we develop rate of convergence results by assuming that the step is a truncated Newton method. Our development is mainly geometrical; this approach allows the development of a convergence theory without any linear independence assumptions.Work supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38.Work supported in part by the National Science Foundation grant DMS-8803206 and by the Air Force Office of Scientific Research grant AFSOR-860080.     

The empirical behavior of sampling methods for stochastic programming

We investigate the quality of solutions obtained from sample-average approximations to two-stage stochastic linear programs with recourse. We use a recently developed software tool executing on a computational grid to solve many large instances of these problems, allowing us to obtain high-quality solutions and to verify optimality and near-optimality of the computed solutions in various ways. Research supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38, and by the National Science Foundation under Grant 9726385. Research supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38, and by the National Science Foundation under Grant DMS-0073770. Research supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38, and by the National Science Foundation under Grants 9726385 and 0082065.     

The equivalence of strict convexity and injectivity of the gradient in bounded level sets

It is shown that Lipschitzian functions are strictly convex if and only if their generalized gradients are disjoint at distinct interior points of a given bounded level set.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, US Department of Energy, under Contract W-31-109-Eng-38.     

Interior point methods for optimal control of discrete time systems

We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discrete-time optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete-time linear-quadratic regulator problem with mixed state/control constraints and show how they can be efficiently-incorporated into an inexact sequential quadratic programming algorithm for nonlinear problems. The key to the efficiency of the interior-point method is the narrow-banded structure of the coefficient matrix which is factorized at each iteration.This research was supported by the Applied Mathematical Sciences Subprogram of the Office of Energy Research, US Department of Energy, under Contract W-31-109-Eng-38.     

A domain decomposition algorithm for elliptic problems in three dimensions

Summary Most domain decomposition algorithms have been developed for problems in two dimensions. One reason for this is the difficulty in devising a satisfactory, easy-to-implement, robust method of providing global communication of information for problems in three dimensions. Several methods that work well in two dimension do not perform satisfactorily in three dimensions.A new iterative substructuring algorithm for three dimensions is proposed. It is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain. The condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains. The new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions.This work was supported in part by the National Science Foundation under grant NSF-CCR-8903003 and by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Constraint identification and algorithm stabilization for degenerate nonlinear programs

 In the vicinity of a solution of a nonlinear programming problem at which both strict complementarity and linear independence of the active constraints may fail to hold, we describe a technique for distinguishing weakly active from strongly active constraints. We show that this information can be used to modify the sequential quadratic programming algorithm so that it exhibits superlinear convergence to the solution under assumptions weaker than those made in previous analyses. Received: December 18, 2000 / Accepted: January 14, 2002 Published online: September 27, 2002 RID="★" ID="★" Research supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38. Key words. nonlinear programming problems – degeneracy – active constraint identification – sequential quadratic programming     

Multigrid and multilevel methods for quadratic spline collocation

Multigrid methods are developed and analyzed for quadratic spline collocation equations arising from the discretization of one-dimensional second-order differential equations. The rate of convergence of the two-grid method integrated with a damped Richardson relaxation scheme as smoother is shown to be faster than 1/2, independently of the step-size. The additive multilevel versions of the algorithms are also analyzed. The development of quadratic spline collocation multigrid methods is extended to two-dimensional elliptic partial differential equations. Multigrid methods for quadratic spline collocation methods are not straightforward: because the basis functions used with quadratic spline collocation are not nodal basis functions, the design of efficient restriction and extension operators is nontrivial. Experimental results, with V-cycle and full multigrid, indicate that suitably chosen multigrid iteration is a very efficient solver for the quadratic spline collocation equations. Supported by Communications and Information Technology Ontario (CITO), Canada. Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Computing mountain passes and transition states

The mountain-pass theorem guarantees the existence of a critical point on a path that connects two points separated by a sufficiently high barrier. We propose the elastic string algorithm for computing mountain passes in finite-dimensional problems and analyze the convergence properties and numerical performance of this algorithm for benchmark problems in chemistry and discretizations of infinite-dimensional variational problems. We show that any limit point of the elastic string algorithm is a path that crosses a critical point at which the Hessian matrix is not positive definite.This work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contract W-31-109-Eng-38.