How can I solve a minimization with 3 linear constraints?

I don't understand what kind of function I can use to solve the problem on photo. It's a problem with Matrix and vector, x is a vector, also E, q_B and 1 are so, while V is a Matrix. G is a scalar number which changes (but with a for loop I can do so). z and VaR are numbers.

Answer 1

[Torsten]

Use MATLAB's "fmincon" to solve.

Best wishes

Torsten.

[Camilla Lincetto]

But in the last constraint I can't esplicate the variable x, so I need to use the nonlinear option, in which I should also compute nonlinear inequality constraint. But I don't have this, so how can I solve this?

[Torsten]

Call fmincon as

x = fmincon(@myfun,x0,[],[],Aeq,beq,[],[],@(x)mycon(x,z,qb,V,a,R))

after defining Aeq and beq according to your linear constraints.

Then define mycon as

function [c,ceq]=mycon(x,z,qb,V,a,R)
c=[];
E=ones(size(x));
ceq=z*sqrt((x+qb)'*V*(x+qb))-(x+qb)'*E-V*a*R;

Best wishes

Torsten.

Answer 2

[Johan Löfberg]

The formulation here looks like a bad idea. The last equality is a nonconvex equality constraints, which thus can be tricky for many solver. However, from the context, it is clear that you can relax the equality to <= (If you can obtain a lower VaR than the one specified, great, but the optimal solution will be tight, i.e., equality will hold). With an <= instead, the constraint is convex.

Using fmincon might work, but there are many solvers specialized to this sort of problems (it is a special case of second-order cone programming), such as mosek, gurobi, cplex, sdpt3,sedumi,...

In addition to that, you have modelling tools which can help you to interface these solvers, such as cvx and YALMIP.

Here is the YALMIP code to solve the problem

x = sdpvar(n,1);
Objective = x'*V*x;
Constraints = [x'*E == G, sum(x)==0, z*norm(chol(V)*(x+qb))-(x+qb)'*E <= VaR];
optimize(Constraints,Objective)
value(x)

If you have installed a solver specialized for these models, it will be used. Otherwise it will resort to fmincon.