How to Solve Linear Complementarity Problem (LCP) with MATLAB ?

2025 6 9
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I want to use MATLAB for solving Linear Complementarity Problem (LCP) as defined in the following:
what optimization function (e.g., fmincon,lsqnonneg,etc.) should I use, or someone could give me some example.

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Matt J
Matt J 2025-6-9
编辑：Matt J 2025-6-9
You don't appear to have an objective function. Are you just trying to find an arbitrary (ω, z) satisfying these (in)equalities?

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Matt J
Matt J 2025-6-9
编辑：Matt J 2025-6-9
Ran in:

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Ran in:
You could do it with quadprog, but you may as well just deploy it as an optimproblem.
n=5; q=-rand(n,1); M=rand(n); M=(M+M')/2;
smallTolerance=1e-6;
z=optimvar('z',n,'Lower',0);
prob=optimproblem('Objective', z'*M*z + z'*q, "Constraints", M*z+q>=0);
[sol,fval]=solve(prob);
Solving problem using quadprog. Minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.
fval
fval = 7.2164e-16
assert(fval<=smallTolerance, "No solution found")
eig(M)', %Not PSD
ans = 1×5
-0.3846 -0.2232 0.0577 0.4325 2.3596
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zOptimal=sol.z %Solutions
zOptimal = 5×1
0.0000 0.8317 0.0000 0.0000 0.0000
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omegaOptimal=M*zOptimal+q
omegaOptimal = 5×1
0.1923 0 0.1923 0.3776 0.6387
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Torsten
Torsten 2025-6-9
编辑：Torsten 2025-6-9
Which solver will be invoked by "solve" ?
Will "quadprog" work if M is not positive-semidefinite ?
Trying
https://uk.mathworks.com/matlabcentral/fileexchange/41485-a-pivoting-algorithm-solving-linear-complementarity-problems
might be an alternative.
Matt J
Matt J 2025-6-9
Quadprog will "work", even if M is not psd, but of couse it might then not find the global solution. I've modified my answer with an example where it succeeds, in spite of this.
Torsten
Torsten 2025-6-9
Don't you have to use
n=5; q=-rand(n,1); M=rand(n); Msym=(M+M')/2;
smallTolerance=1e-6;
z=optimvar('z',n,'Lower',0);
prob=optimproblem('Objective', z'*Msym*z + z'*q, "Constraints", M*z+q>=0);
[sol,fval]=solve(prob);
fval
assert(fval<=smallTolerance, "No solution found")
eig(Msym)', %Not PSD
zOptimal=sol.z %Solutions
omegaOptimal=M*zOptimal+q
for M being a general nxn matrix ?
Chuguang Pan
Chuguang Pan 2025-6-10
@Matt J. Thanks for your detailed reply, your answer is helpful.

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