Quadratic Programming and Cone Programming

Solve problems with quadratic objectives and linear constraints or with conic constraints

Before you begin to solve an optimization problem, you must choose the appropriate approach: problem-based or solver-based. For details, see First Choose Problem-Based or Solver-Based Approach.

For the problem-based approach, create problem variables, and then represent the objective function and constraints in terms of these symbolic variables. For the problem-based steps to take, see Problem-Based Optimization Workflow. To solve the resulting problem, use solve.

For the solver-based steps to take, including defining the objective function and constraints, and choosing the appropriate solver, see Solver-Based Optimization Problem Setup. To solve the resulting problem, use quadprog or coneprog.

Functions

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Solve and Analyze, Problem-Based

`evaluate`	Evaluate optimization expression or objectives and constraints in problem
`infeasibility`	Constraint violation at a point
`optimproblem`	Create optimization problem
`optimvar`	Create optimization variables
`solve`	Solve optimization problem or equation problem

Solve Problems, Solver-Based

`coneprog`	Second-order cone programming solver (Since R2020b)
`optim.coder.infbound`	Infinite bound support for code generation (Since R2022b)
`optimwarmstart`	Create warm start object (Since R2021a)
`quadprog`	Quadratic programming
`secondordercone`	Create second-order cone constraint (Since R2020b)

Live Editor Tasks

Optimize

Optimize or solve equations in the Live Editor (Since R2020b)

Objects

SecondOrderConeConstraint Second-order cone constraint object (Since R2020b)

Topics

Problem-Based Quadratic Programming

Quadratic Programming with Bound Constraints: Problem-Based
Shows how to solve a problem-based quadratic programming problem with bound constraints using different algorithms.
Large Sparse Quadratic Program, Problem-Based
Shows how to solve a large sparse quadratic program using the problem-based approach.
Bound-Constrained Quadratic Programming, Problem-Based
Example showing large-scale problem-based quadratic programming.
Quadratic Programming for Portfolio Optimization, Problem-Based
Example showing problem-based quadratic programming on a basic portfolio model.
Diversify Portfolios Using Optimization Toolbox
This example shows three techniques of asset diversification in a portfolio using optimization functions.

Solver-Based Quadratic Programming

Quadratic Minimization with Bound Constraints
Example of quadratic programming with bound constraints and various options.
Quadratic Programming with Many Linear Constraints
This example shows the benefit of the active-set algorithm on problems with many linear constraints.
Warm Start quadprog
Shows that warm start can be effective in a large quadratic program.
Warm Start Best Practices
Describes how best to use warm start for speeding repeated solutions.
Quadratic Minimization with Dense, Structured Hessian
Example showing how to save memory in a structured quadratic program.
Large Sparse Quadratic Program with Interior Point Algorithm
Example showing how to save memory in a quadratic program by using a sparse quadratic matrix.
Bound-Constrained Quadratic Programming, Solver-Based
Example showing solver-based large-scale quadratic programming.
Quadratic Programming for Portfolio Optimization Problems, Solver-Based
Example showing solver-based quadratic programming on a basic portfolio model.

Problem-Based Second-Order Cone Programming

Minimize Energy of Piecewise Linear Mass-Spring System Using Cone Programming, Problem-Based
Presents a problem-based example of cone programming.
Discretized Optimal Trajectory, Problem-Based
This example shows how to solve a discretized optimal trajectory problem using the problem-based approach.
Compare Speeds of coneprog Algorithms
This section gives timing information for a sequence of cone programming problems using various LinearSolver option settings.
Write Constraints for Problem-Based Cone Programming
Requirements for solve to use coneprog for problem solution.

Solver-Based Second-Order Cone Programming

Minimize Energy of Piecewise Linear Mass-Spring System Using Cone Programming, Solver-Based
Solve a mechanical mass-spring problem using cone programming.
Convert Quadratic Constraints to Second-Order Cone Constraints
Convert quadratic constraints into coneprog form.
Convert Quadratic Programming Problem to Second-Order Cone Program
Convert a quadratic programming problem to a second-order cone problem.

Code Generation

Code Generation for quadprog Background
Prerequisites to generate C code for quadratic optimization.
Generate Code for quadprog
Learn the basics of code generation for the quadprog optimization solver.
Warm Start Best Practices
Describes how best to use warm start for speeding repeated solutions.
Optimization Code Generation for Real-Time Applications
Explore techniques for handling real-time requirements in generated code.

Problem-Based Algorithms

Problem-Based Optimization Algorithms
Learn how the optimization functions and objects solve optimization problems.
Write Constraints for Problem-Based Cone Programming
Requirements for solve to use coneprog for problem solution.
Supported Operations for Optimization Variables and Expressions
Explore the supported mathematical and indexing operations for optimization variables and expressions.

Algorithms and Options

Quadratic Programming Algorithms
Minimizing a quadratic objective function in n dimensions with only linear and bound constraints.
Second-Order Cone Programming Algorithm
Description of the underlying algorithm.
Optimization Options Reference
Explore optimization options.