Optimization Toolbox 4.1

Product Description

• Introduction and Key Features
• Defining, Solving, and Assessing Optimization Problems
• Nonlinear Optimization and Multi-Objective Optimization
• Nonlinear Least-Squares, Data Fitting, and Nonlinear Equations
• Quadratic, Linear, and Binary Integer Programming
• Solving Optimization Problems Using Parallel Computing

Nonlinear Optimization and Multi-Objective Optimization

Unconstrained Nonlinear Optimization

Optimization Toolbox uses three methods to solve unconstrained nonlinear minimization problems: Quasi-Newton, Nelder-Mead, and trust-region.

The Quasi-Newton method uses a mixed quadratic and cubic line search procedure and the BFGS formula for updating the approximation of the Hessian matrix.

Nelder-Mead is a direct-search method that uses only function values (does not require derivatives) and handles non-smooth objective functions.

Trust-region is used for large-scale problems where sparsity or structure can be exploited.

Constrained Nonlinear Optimization

Constrained nonlinear optimization problems are composed of nonlinear objective functions and may be subject to linear and nonlinear constraints. Optimization Toolbox uses three methods to solve these problems: trust-region, active-set sequential quadratic programming, and interior-point.

Trust-region is used for bound constrained problems or linear equalities only.

Active-set sequential quadratic programming is used for general nonlinear optimization.

Interior-point is used for general nonlinear optimization. It is especially valuable for large-scale problems. It is based on a barrier function, and optionally keeps all iterates strictly feasible with respect to bounds during the optimization run.

The trust-region method, which is based on an interior-reflective Newton method, allows you to estimate Hessians with known sparsity structure using sparse finite differences of gradients. Additionally, the trust-region and interior-point methods enable you to calculate Hessian-times-vector products in a function without having to form the Hessian matrix explicitly. You can also adjust the bandwidth of the preconditioner used in the Newton iteration.

Optimization Toolbox also includes an interface to Ziena Optimization's KNITRO® libraries for solving constrained and unconstrained nonlinear optimization problems. Use of the interface requires an installation of the KNITRO libraries, which can be purchased separately from Ziena Optimization, Inc.

Multi-objective Optimization

Multi-objective optimization is concerned with the minimization of multiple objective functions that are subject to a set of con straints. Optimization Toolbox provides functions for solving two formulations of multi-objective optimization problems: goal attainment and minimax.

The goal attainment problem involves reducing the value of a linear or nonlinear vector function to attain the goal values given in a goal vector. The relative importance of the goals is indicated using a weight vector. The goal attainment problem may also be subject to linear and nonlinear constraints.

The minimax problem involves minimizing the worst-case value of a set of multivariate functions, possibly subject to linear and non linear constraints.

Optimization Toolbox transforms both types of multi-objective problems into standard constrained optimization problems and then solves them using a sequential quadratic programming approach.