Mosek - Computer Sciences Department

MOSEK 6The original problem is:∑d ii x iminimizesubject to a t x ≤ bx i ∈ [l i , u i ], l i > 0, d i ≥ 0, i = 1, 2, ..., n(1.9)where x ∈ R n is the decision variable, d, a, l, u ∈ R n parameters, and b ∈ R a scalar parameter. The originalmodel (1.9) can be written in GAMS using the equations:defobj.. sum(n, d(n)/x(n)) =E= obj;e1.. sum(n, a(n)*x(n)) =L= b;Model orig /defobj, e1/;x.lo(n) = l(n);x.up(n) = u(n);We can write an equivalent NLP formulation, replacing the objective function and adding another constraint:∑minimizei d it isubject to a t x ≤ b(1.10)2t i x i ≥ 2, i = 1, ..., nx ∈ [l, u], l > 0, d i ≥ 0where t ∈ R n is a new decision variable. The GAMS formulation of this NLP (model cnlp) is:defobjc.. sum(n, d(n)*t(n)) =E= obj;e1.. sum(n, a(n)*x(n)) =L= b;conenlp(n).. 2*t(n)*x(n) =G= 2;Model cnlp /defobjc, e1, conenlp/;x.lo(n) = l(n);x.up(n) = u(n);We can change the equality to an inequality since the parameter d i ≥ 0 and we are dealing with a minimizationproblem. Also, note that the constraint conenlp(n) is almost in rotated conic form. If we introduce a variablez ∈ R n , z i = √ 2, then we can reformulate the problem using conic constraints as:The GAMS formulation using conic equations =C= is:∑minimizei d it isubject to a t x ≤ bz i = √ 22t i x i ≥ zi 2 , i = 1, ..., nx ∈ [l, u], l > 0, d i ≥ 0(1.11)defobjc.. sum(n, d(n)*t(n)) =E= obj;e1.. sum(n, a(n)*x(n)) =L= b;e2(n).. z(n) =E= sqrt(2);cone(n).. x(n) + t(n) =C= z(n);Model clp /defobjc, e1, e2, cone/;x.lo(n) = l(n);x.up(n) = u(n);Note that this formulation is a linear program in GAMS, although the constraints cone(n)... represent thenonlinear rotated quadratic cone constraint.The complete model is listed below: