What Is Optimization Toolbox?

Large-Scale ExamplesGenerally speaking, the large-scale optimization methods preserve structureand sparsity, using exact derivative information wherever possible. To solvethe large-scale problems efficiently, some problem formulations are restricted(such as only solving overdetermined linear or nonlinear systems), or requireadditional information (e.g., the nonlinear minimization algorithm requiresthat the gradient be computed in the user-supplied function).This section summarizes the kinds of problems covered by large-scale methodsand provides examples.Problems Covered by Large-Scale MethodsThis section describes how to formulate problems for functions that uselarge-scale methods. It is important to keep in mind that there are somerestrictions on the types of problems covered by large-scale methods. Forexample, the function fmincon cannot use large-scale methods when thefeasible region is defined by either of the following:• Nonlinear equality or inequality constraints• Both upper- or lower-bound constraints and equality constraintsWhen a function is unable to solve a problem using large-scale methods, itreverts to medium-scale methods.Formulating Problems with Large-Scale MethodsThefollowingtablesummarizeshowto set up problems for large-scalemethods and provide the necessary input for the optimization functions. Foreach function, the second column of the table describes how to formulatethe problem and the third column describes what additional information isneeded for the large-scale algorithms. For fminunc and fmincon, the gradientmust be computed along with the objective in the user-supplied function (thegradient is not required for the medium-scale algorithms).Since these methods can also be used on small- to medium-scale problemsthat are not necessarily sparse, the lastcolumnofthetableemphasizeswhatconditions are needed for large-scale problems to run efficiently withoutexceeding your computer system’s memory capabilities, e.g., the linearconstraint matrices should be sparse. For smaller problems the conditionsin the last column are unnecessary.2-41