which solver can solve this model?

[覃岭]

my model has a following constraint:

A'XAY==C,

in which A is a symmetrical matrix with 0, 1 constants, 

X is a diag matrix of 0-1 variables, 

Y is a matrix of real variables,

C is a constant matrix.

 

obviously nonlinear terms are contained in this constraint,

is there any tricks to convert it into linear constraints, or any solver can solve it directly?

[Johan Löfberg]

Yes, products between continuous variables and binaries can easily be converted to a pure MILP model by using the fact that x*y is equal to y if x is 1 and equal to 0 if x is zero, i.e., a purely logical property.

YALMIP implements this for general polynomial expressions in binmodel. Hence, your model can be solved with any MILP solver and YALMIP as

A = randn(3,3);
X = diag(binvar(3,1));
Y = sdpvar(3,3,'full');
knownY = randn(3);
C = A'*A*knownY;

[AXAY, milpmodel] = binmodel(A'*X*A*Y,[-10 <= Y(:)<=10]);

solvesdp([AXAY == C, milpmodel])

[knownY double(Y)]

Note that known bounds on the continuous variable is required.

[Johan Löfberg]

BTW, if binmodel crashes, comment out line 50 in binmodel.m. I seem to have introduced a bug in the latest release

[覃岭]

thanks, johan, your answer is very good. And I have another way to deal with it:

A'xAy ==C can be extracted as a set of constraints with polynomial in left hand side, for first constraint:

x1*y1 + ... + xn*yn == c, where variables xi∈{0,1}, yi∈R,and constant b∈R

    I try to add a series of auxiliary variables zi:

    z1+...+zn==c,

    for i=1,...n

    yi - M*(1-xi) <= zi <= yi + M*(1-xi)

    -M*xi <= zi <=  M*xi

[Johan Löfberg]

That's basically the model binmodel derives automatically for you.