Solver-Based Optimization in MATLAB

Define and solve optimization and least-squares problems and systems of nonlinear equations. Use the Optimize Live Editor task to guide you through this workflow.

A screenshot of the Optimize Live Editor task in MATLAB software, specifically configured for multiobjective optimization using the paretosearch solver.

1. Group the optimization variables into a single vector \( x \). Write the objective and constraints in terms of \( x \).

Objective Type	Mathematical Form	Example
Linear	\( f^{T} x \)	`f = [-1 0 -5];`
Quadratic	\( x^{T}H x + f^{T} x \)	`H = [5 1 0; 1 3 0; 0 0 0];`
Least Squares	\( \\| Cx - d \\|_2 \) \( \sum F_i(x)^2 \)	`C = [7 8 10; 1 3 4; 2 5 7];` `d = [2; 1; 1.5];` `function F = myF(x)` `F(1) = f1(x);` `F(2) = f2(x);` `end`
General	\( f(x) \)	`function objval = fobj(x)` `objval = 3*(x(1)-x(2))^4;` `end`

Constraint Type	Mathematical Form	Example
Bound	\( l \leq x \leq u \)	`lb = zeros(n,1);` `ub = 5*ones(n,1);`
Linear	\( A x \leq b \) \( A_{eq} x = b_{eq} \)	`A = [1 0 1; 0 -2 1];` `b = [4; 2];` `Aeq = [1 0 2];` `beq = 1;`
Second-Order Cone	\( \\| A_{sc} x - b_{sc} \\| \leq d_{sc} x - \gamma \)	`A = diag([1,1/2,0]);` `b = zeros(3,1);` `d = [0;0;1];` `gamma = 0;` `socConstraints =` `secondordercone(A,b,d,gamma);`
General	\( c(x) \leq 0 \) \( c_{eq}(x) = 0 \)	`function[c,ceq] = nlcons(x)` `c(1) = x(1).^2 + x(2).^2 - 1;` `c(2) = x(1)*x(3) - 5;` `ceq = [];` `end`
Integer	\( x_j \in \mathbb{Z}^n \)	`intcon = [1 2]`

2. Choose a solver matching the types of objective and constraints.

Solvers in Optimization Toolbox™ use derivatives, are usually faster, and scale to large problems. Solvers in Global Optimization Toolbox (italic) and MATLAB (*) do not use derivatives and search for global minima.

Constraint Type	Objective Type
Constraint Type	Linear	Quadratic	Least Squares	General Smooth	General Nonsmooth	Multiobjective
None		`quadprog`	`lsqcurvefit` `lsqnonlin` `mldivide`	*`fminsearch``fminunc`**	*`fminsearch``patternsearch ga particleswarm simulannealbnd`**	`fgoalattain``fminimax paretosearch gamultiobj`
Bound	`linprog`	`quadprog`	`lsqcurvefit lsqnonlin lsqnonneg lsqlin`	`fmincon`	`surrogateoptpatternsearch ga fminbnd* particleswarm simulannealbnd`	`fgoalattain fminimax paretosearch gamultiobj`
Linear	`linprog`	`quadprog`	`lsqlin`	`fmincon`	`patternsearch ga` *surrogateopt*	`fgoalattain fminimax paretosearch gamultiobj`
Second-Order Cone	`coneprog`	`coneprog`
General Smooth	`fmincon`	`fmincon`	`fmincon`	`fmincon`	`patternsearch` `ga``surrogateopt`	`fgoalattain``fminimax paretosearch gamultiobj`
General Nonsmooth	`patternsearch ga` `surrogateopt`	`patternsearch ga` `surrogateopt`	`patternsearch ga` `surrogateopt`	`patternsearch ga` `surrogateopt`	`patternsearch ga surrogateopt`	`paretosearch gamultiobj`
Integer	`intlinprog`				`ga surrogateopt`

3. Define initial point if required and options if desired. Call solver and obtain solution.

Initial Point

Examples:

x0 = lb + 0.5*(ub-lb)
x0 = zeros(n,1)

Options

Use optimoptions to set stopping criteria, plot functions, initial population, and more.

Examples:

opts = optimoptions('fmincon','Display','iter')

Solve

Examples:

[x,fval] = fmincon(@fobj,x0,A,b,Aeq,beq,lb,ub,@nlcons,opts)
[x,fval,eflag] = ga(@fobj,nvars)
x = lsqlin(C,d,A,b,[],[],lb)

Do More
Interpret and improve results Pass extra parameters to functions Solver comparison table and example Solve systems of nonlinear equations Search for global minima on smooth problems

Solver-Based Optimization in MATLAB

Define and solve optimization and least-squares problems and systems of nonlinear equations. Use the Optimize Live Editor task to guide you through this workflow.

1. Group the optimization variables into a single vector \( x \). Write the objective and constraints in terms of \( x \).

2. Choose a solver matching the types of objective and constraints.

3. Define initial point if required and options if desired. Call solver and obtain solution.

Learn more: