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I want find the minima of a (multivariable) function under a constraint which has to be fulfilled on a whole interval, let's say $$ \nabla f (\underline x) = 0 \ \\ \ c(\underline x,s)\geq0\ \forall s\in [0,1]. $$ How do I implement such a condition into the Minimize[{f[x1,x2,...,xn],c[x1,...,xn,s]>=0 ?},{x1,x2,...,xn}] function? Thanks in advance!

Edit: Ok, small mistake. I wanted a condition to be an inequality. If I just change that in the example proposed it is stated that this are not valid constraints:

f[x_, y_] := x^2 + y^2;
c[x_, y_, s_] = 2 x + 3 y + s;
NMinimize[{f[x, y], c[x, y, s] >= 0, 1 >= s >= 0}, {x, y}]
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    – bbgodfrey
    Commented Jan 31, 2015 at 16:34

2 Answers 2

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Here's a two dimensional example:

f[x_, y_] := x^2 + y^2;
c[x_, y_, s_] = 2 x + 3 y + s;
Minimize[{f[x, y], c[x, y, s] == 0, 1 >= s >= 0}, {x, y}]

This gives an answer that depends on the value of s, as is plausible. For your revised/edited problem:

Minimize[{f[x, y], c[x, y, s] >= 0, 1 >= s >= 0}, {x, y, s}]
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  • $\begingroup$ But doesn't that just imply that c[x,y,s]>=0 for just one special s and not for all? I wanted to enforce the constraint on the whole interval. $\endgroup$
    – Malte B
    Commented Feb 3, 2015 at 14:38
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This belongs to the class of Semi-Infinite Programming problems for which ad-hoc algorithms must be used. The most intuitive one (discretization) involves building a finite grid on the interval and imposing the constraint on the grid points only, thus obtaining a classical constrained optimization problem with n constraints. One solves a sequence of such discretized problems on increasingly fine grids until a suitable convergence criterion is satisfied. You can check out the Wikipedia page for a list of references on this and other approaches: https://en.m.wikipedia.org/wiki/Semi-infinite_programming

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