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), second-order cone programming (SOCP), 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 automatic differentiation of objective and constraint functions for faster and more accurate solutions.
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 enables you to find optimal solutions in applications such as portfolio optimization, energy management and trading, and production planning.
Get Started:
Modeling
Transform a problem description into a mathematical form by defining variables, objectives, and constraints, so that it can be solved with optimization techniques.
Problem-Based Optimization
Write objectives and constraints with expressions of optimization variables. Solve faster and more robustly with automatic differentiation on the nonlinear expressions. Apply an automatically selected solver. Interactively create and solve the problem with the Optimize Live Editor task and then generate code for sharing or use in your application.
Solver-Based Optimization
Write nonlinear objectives and constraints using functions; write linear objectives and constraints using coefficient matrices. Interactively create and solve the problem with the Optimize Live Editor task and then generate code for sharing or use in your application.
Choosing a Solver
Use the Optimize Live Editor task with the problem-based or solver-based approach to help choose a solver suitable for the type of problem.
Setting Options
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 exit messages, optimality measures, and the iterative display to assess the solution. Improve performance on nonlinear problems by using automatic differentiation, supplying gradients, or using parallel computing to estimate gradients.
Monitoring solver progress with the iterative display.
Solvers
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.
Applications
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.
Feasible region and optimal solution of a linear program.
Quadratic and Second-Order Cone Programming Solvers
Apply interior-point, active-set, or trust-region-reflective algorithms to solve quadratic programs. Apply interior-point methods to solve second-order cone programs.
Feasible region and optimal solution of a quadratic program.
Applications
Use linear programming on problems such as resource allocation, production planning, blending, and investment planning. Use quadratic and second-order cone programming on problems such as design optimization, portfolio optimization, and control of hydroelectric dams.
Optimal control strategy found with quadratic programming.
Solvers
Solve mixed-integer linear programming problems using the branch and bound algorithm, which includes preprocessing, heuristics for generating feasible points, and cutting planes.
Applying the branch and bound algorithm.
Mixed-Integer Linear Programming-Based Algorithms
Use the mixed-integer linear programming solver to build special-purpose algorithms.
The shortest tour visiting each city only once.
Applications
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.
Schedule for two generators under varying electricity prices.
Solvers
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.
Pareto front computed using the fgoalattain function.
Applications
Use multiobjective optimization when tradeoffs are required for conflicting objectives. Examples are weight and strength in structural design and risk and return in portfolio optimization.
Magnitude response for initial and optimized filter coefficients.
Solvers
Apply Levenberg-Marquardt, trust-region, active-set, or interior-point algorithms.
Comparison of local and global approaches.
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.
Recovering a blurred image by solving a linear least-squares problem.
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, linear, and nonlinear constraints.
Fitting a circular path to the Lorenz system of ordinary differential equations.
MATLAB Compiler Support
Use MATLAB Compiler™ and MATLAB Compiler SDK™ to deploy MATLAB® optimization models as standalone executables, web apps, C/C++ shared libraries, Microsoft® .NET assemblies, Java® classes, and Python® packages.
App that computes an optimal power generation schedule.
Code Generation
Generate portable and readable C or C++ code to solve optimization problems using MATLAB Coder™. Compile the generated code for any hardware, including embedded systems.
MATLAB Coder report for a trajectory optimization function.
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