Solve linear, quadratic, integer, and nonlinear optimization problems
Optimization Toolbox™ provides functions for finding parameters that minimize or maximize objectives while satisfying constraints. The toolbox includes solvers for linear programming (LP), mixed-integer linear programming (MILP), quadratic programming (QP), nonlinear programming (NLP), constrained linear least squares, nonlinear least squares, and nonlinear equations. You can define your optimization problem with functions and matrices or by specifying variable expressions that reflect the underlying mathematics.
You can use the toolbox solvers to find optimal solutions to continuous and discrete problems, perform tradeoff analyses, and incorporate optimization methods into algorithms and applications. The toolbox lets you perform design optimization tasks, including parameter estimation, component selection, and parameter tuning. It can be used to find optimal solutions in applications such as portfolio optimization, resource allocation, and production planning and scheduling.
Transform a problem description into a mathematical form by defining optimization variables, an objective, and constraints, so that it can be solved with optimization techniques.
Write the objective and constraints in MATLAB using expressions that reflect the mathematical form. Build these with optimization variables, mathematical operators, matrix operations, and functions of the variables. Apply an automatically selected solver.
Write the objective and constraints in MATLAB using functions when nonlinear and coefficient matrices when linear. Build these with MATLAB matrix operations, mathematical operators, and functions. Apply an appropriate solver.
Choosing a Solver
Solve using a nonlinear, quadratic, linear, mixed-integer, least-squares, or multiobjective solver depending on the types of objectives and constraints in the optimization problem. The solver is automatically selected in the problem-based approach.
Set optimization options to tune the optimization process, for example, to choose the optimization algorithm used by the solver, or to set termination conditions. Set options to monitor and plot optimization solver progress.
Reviewing and Improving Results
Review the optimality measures and iterative display to assess the solution. Use the links in the exit messages to learn more. Improve performance on nonlinear problems by supplying gradients or using parallel computing to estimate gradients.
Apply quasi-Newton, trust-region, or Nelder-Mead Simplex algorithms to solve unconstrained problems. Apply interior-point, sequential-quadratic-programming (SQP), or trust-region-reflective algorithms to solve constrained problems.
Use nonlinear optimization for estimating and tuning parameters, finding optimal designs, computing optimal trajectories, constructing robust portfolios, and other applications where there is a nonlinear relationship between variables.
Linear Programming Solvers
Apply dual-simplex or interior-point algorithms to solve linear programs.
Quadratic Programming Solvers
Apply interior-point, active-set, or trust-region-reflective algorithms to solve quadratic programs.
Use linear programming on problems such as resource allocation, production planning, blending, and investment planning. Use quadratic programming on problems such as design optimization, portfolio optimization, and control of hydroelectric dams.
Solve mixed-integer linear programming problems using the branch and bound algorithm, which includes preprocessing, heuristics for generating feasible points, and cutting planes.
Mixed-Integer Linear Programming-Based Algorithms
Use the mixed-integer linear programming solver to build special-purpose algorithms.
Model with integer variables when there are on/off decisions or logical constraints as well as when variable values must be integral. Routing, scheduling, planning, assignment, and capital budgeting problems are typical applications.
Formulate problems as either goal-attainment or minimax. Use goal-attainment when there are optionally weighted goal values for each of the objectives. Use minimax to minimize the worst-case value of a set of objective functions.
Use multiobjective optimization when tradeoffs are required for two or more conflicting objectives. Examples are weight and strength in structural design and risk and return in portfolio optimization.
Apply Levenberg-Marquardt, trust-region, active-set, or interior-point algorithms.
Linear Least-Squares Applications
Use linear least-squares solvers to fit a linear model to acquired data or to solve a system of linear equations, including when the parameters are subject to bound and linear constraints.
Nonlinear Least-Squares Applications
Use nonlinear least-squares solvers to fit a nonlinear model to acquired data or to solve a system of nonlinear equations, including when the parameters are subject to bound constraints.
Generate C/C++ code to solve quadratic programming problems with
quadprog (requires MATLAB Coder)
Generate C/C++ code to solve nonlinear constrained optimization problems with
fmincon (requires MATLAB Coder)
Define nonlinear optimization objective functions or constraints using elementary functions of optimization variables
Quadratic Programming and Linear Least Squares
Solve dense problems quickly and accurately using the active-set algorithm
Mixed-Integer Linear Programming
Obtain feasible points faster with additional heuristics
Infeasibility Analysis Example
Identify conflicting linear constraints by finding irreducible infeasible or maximal feasible subsets