Problem-Based Optimization with Optimization Toolbox

Following the problem-based workflow, first create an optimization problem with optimproblem to hold the objective, constraints, and associated variables. Create an eqnproblem when solving a system of nonlinear equations.

Examples:

assignmentProb = optimproblem
responseProb = optimproblem
initialStateProb = eqnproblem

Create optimization variables with optimvar. Set display name and optional dimensions, bounds, and type. Index with integers or character strings.

Examples:

x = optimvar("x");
y = optimvar("y");

employees = ["a","b","c"];
tasks = ["t1","t2","t3"];
assign = optimvar("assign",employees,tasks,"LowerBound",0,"UpperBound",1,"Type","integer")

Directly specify an OptimizationExpression with supported operations.

Examples:

response = -3*(y - x.^3 - x).^2 - (x - 4/3).^2;
totalCost = sum(sum(cost.*assign));
sumByEmployee = sum(assign,2);
sumByTask = sum(assign,1);

Use any MATLAB^® function by converting it to an optimization expression with fcn2optimexpr.

Examples:

a = 4;
xyfcn = @(x,y,a)gamma(y)*a*x.^2;
xyexpr = fcn2optimexpr(xyfcn,x,y,a);

Examples:

responseProb.ObjectiveSense = "maximize";
responseProb.Objective = response;

assignmentProb.ObjectiveSense = "minimize";
assignmentProb.Objective = totalCost;

Combine OptimizationExpressions with a relational operator to specify an OptimizationConstraint or an OptimizationEquality. Assign to a problem.

Examples:

responseProb.Constraints.ellipse = x.^2/2 + y.^2/4 <= 1;
responseProb.Constraints.xyconstr = xyexpr >= 1;

assignmentProb.Constraints.oneTaskPerEmployee = sumByTask <= 1;
assignmentProb.Constraints.oneEmployeePerTask = sumByEmployee == 1;

initialStateProb.Equations.eq1 = x*cos(y) + y*sin(x) == 1/2;
initialStateProb.Equations.eq2 = exp(-exp(-(x + y))) == y*(1 + x^2);

Solve the problem, providing an initial point for nonlinear problems. The solve function returns solution values, objectives values, the reason the problem stopped, and more.

Example:

x0.x = 0;
x0.y = 0;
[sol,fval,exitflag] = solve(responseProb,x0)

Solve with optimization options.​

Example:

o = optimoptions(assignProb,"MaxTime",10);
sol = solve(assignmentProb,"Options",o)

Do More​

• Use evaluate and infeasibility to analyze results​
• Interpret and improve results​
• Convert to solver-based form with prob2struct​
• Include derivatives​