Bound-constrained Optimization Problems Research Articles

Mixed Nonlinear Complementarity Problems via Nonlinear Optimization: Numerical Results on Multi-Rigid-Body Contact Problems with Friction

In this work we show that the mixed nonlinear complementarity problem may be formulated as an equivalent nonlinear bound-constrained optimization problem that preserves the smoothness of the original data. One may thus take advantage of existing codes for bound-constrained optimization. This approach is implemented and tested by means of an extensive set of numerical experiments, showing promising results. The mixed nonlinear complementarity problems considered in the tests arise from the discretization of a motion planning problem concerning a set of rigid 3D bodies in contact in the presence of friction. We solve the complementarity problem associated with a single time frame, thus calculating the contact forces and accelerations of the bodies involved.

Optimizing an Empirical Scoring Function for Transmembrane Protein Structure Determination

We examine the problem of transmembrane protein structure determination. Like many questions that arise in biological research, this problem cannot be addressed generally by traditional laboratory experimentation alone. Instead, an approach that integrates experiment and computation is required. We formulate the transmembrane protein structure determination problem as a bound-constrained optimization problem using a special empirical scoring function, called Bundler, as the objective function. In this paper, we describe the optimization problem and its mathematical properties, and we examine results obtained using two different derivative-free optimization algorithms.

An Overview of First-Order Model Management for Engineering Optimization

First-order approximation/model management optimization (AMMO) is a rigorous methodology for solving high-fidelity optimization problems with minimal expense in high-fidelity function and derivative evaluation. AMMO is a general approach that is applicable to any derivative based optimization algorithm and any combination of high-fidelity and low-fidelity models. This paper gives an overview of the principles that underlie AMMO and puts the method in perspective with other similarly motivated methods. AMMO is first illustrated by an example of a scheme for solving bound-constrained optimization problems. The principles can be easily extrapolated to other optimization algorithms. The applicability to general models is demonstrated on two recent computational studies of aerodynamic optimization with AMMO. One study considers variable-resolution models, where the high-fidelity model is provided by solutions on a fine mesh, while the corresponding low-fidelity model is computed by solving the same differential equations on a coarser mesh. The second study uses variable-fidelity physics models, with the high-fidelity model provided by the Navier-Stokes equations and the low-fidelity model—by the Euler equations. Both studies show promising savings in terms of high-fidelity function and derivative evaluations. The overview serves to introduce the reader to the general concept of AMMO and to illustrate the basic principles with current computational results.

Newton's Method for Large Bound-Constrained Optimization Problems

We analyze a trust region version of Newton's method for bound-constrained problems. Our approach relies on the geometry of the feasible set, not on the particular representation in terms of constraints. The convergence theory holds for linearly constrained problems and yields global and superlinear convergence without assuming either strict complementarity or linear independence of the active constraints. We also show that the convergence theory leads to an efficient implementation for large bound-constrained problems.