In this paper we present a modified nearly exact (MNE) method for solving low-rank trust region (LRTR) subproblem. The LRTR subproblem is to minimize a quadratic function, whose Hessian is a positive diagonal matrix plus explicit low-rank update, subject to a Dikin-type ellipsoidal constraint, whose scaling matrix is positive definite and has the similar structure … Read more
We describe version 3.0 of the COPS set of nonlinearly constrained optimization problems. We have added new problems, as well as streamlined and improved most of the problems. We also provide a comparison of the FILTER, KNITRO, LOQO, MINOS, and SNOPT solvers on these problems. Citation Technical Report ANL/MCS-TM-273, Argonne National Laboratory, 02/04. Article Download … Read more
A sequential quadratic programming algorithm for solving nonlinear programming problems is presented. The new feature of the algorithm is related to the definition of the merit function. Instead of using one penalty parameter per iteration and increasing it as the algorithm progresses, we suggest that a new point is to be accepted if it stays … Read more
We examine the importance of optimality measures when benchmarking a set of solvers, and show that scaling requirements lead to a convergence test for nonlinearly constrained optimization solvers that uses a mixture of absolute and relative error measures. We demonstrate that this convergence test is well behaved at any point where the constraints satisfy the … Read more
In this paper we develop an interior-point primal-dual long-step path-following algorithm for convex quadratic programming (CQP) whose search directions are computed by means of an iterative (linear system) solver. We propose a new linear system, which we refer to as the \emph{augmented normal equation} (ANE), to determine the primal-dual search directions. Since the condition number … Read more
Engineers have been using \emph{bilevel decomposition algorithms} to solve certain nonconvex large-scale optimization problems arising in engineering design projects. These algorithms transform the large-scale problem into a bilevel program with one upper-level problem (the master problem) and several lower-level problems (the subproblems). Unfortunately, there is analytical and numerical evidence that some of these commonly used … Read more
An interior-point method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primal-dual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty strictly feasible region. In contrast to previous approaches, the proposed relaxation … Read more
Decomposition algorithms exploit the structure of large-scale optimization problems by breaking them into a set of smaller subproblems and a coordinating master problem. Cutting-plane methods have been extensively used to decompose convex problems. In this paper, however, we focus on certain nonconvex problems arising in engineering. Engineers have been using bilevel decomposition algorithms to tackle … Read more
We establish polynomial-time convergence of infeasible-interior-point methods for conic programs over symmetric cones using a wide neighborhood of the central path. The convergence is shown for a commutative family of search directions used in Schmieta and Alizadeh. These conic programs include linear and semidefinite programs. This extends the work of Rangarajan and Todd, which established … Read more
Stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning. The deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets, time stages and scenarios per time stage. So far models treated by mathematical programming approaches have been limited to simple linear … Read more