# Numerically finding a minimax solution [closed]

I have a rather complex optimization problem of the following general form: $$f(a_1, \dots, a_k) = \min_{(b_1, \dots, b_m) \in \Omega_b} \left\{\max_{(c_1, \dots, c_n) \in \Omega_c} g(a_1, \dots, a_k, b_1, \dots, b_m, c_1, \dots, c_n)\right\}.$$ Or, in vector form, $$f(\mathbf{a}) = \min_{\mathbf{b} \in \Omega_b} \Big\{\max_{\mathbf{c} \in \Omega_c} \, g(\mathbf{a}, \mathbf{b}, \mathbf{c})\Big\}.$$

In other words, I first want to maximize a given function over a number of variables, then find the minimum over a different set of variables, and then evaluate the resulting solution "function" at certain points.

The function $g$ is so messy that I do not expect to find an explicit form for $f(\mathbf{a})$, but I would still like to be able to (numerically) evaluate $f$ at some points, to get an idea of how $f$ behaves.

In terms of Mathematica, the inner maximization can be done numerically with FindMaximum, but this does not work if the other variables are not explicitly instantiated; so a solution of the form FindMinimum[FindMaximum[g[a,b,c], c], b] unfortunately does not work. For given $\mathbf{a}$ I could try to manually build a list of values FindMaximum[g[a,b,c], c] for a whole range of values $\mathbf{b} \in \Omega_{b}$, and take the minimum over these values, but I was wondering: doesn't Mathematica have some cleaner, more direct way to numerically compute minimax solutions?

(In my application, $(k, m, n) = (3, 6, 6)$ and $\Omega_b, \Omega_c$ are continuous subsets of $\mathbb{R}^6$, so this is not a discrete optimization problem where all possible solutions can just be enumerated. Also, I expect $g$ to be reasonably smooth, so I'm looking for a saddle-point solution in $\mathbb{R}^{12}$.)

## closed as off-topic by MarcoB, m_goldberg, corey979, Bob Hanlon, ÖskåFeb 24 '17 at 18:17

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question cannot be answered without additional information. Questions on problems in code must describe the specific problem and include valid code to reproduce it. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included." – MarcoB, m_goldberg, corey979, Bob Hanlon, Öskå
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• Is this question related to any other question on mathematics.stack.exchange ? – andre314 Feb 11 '17 at 1:25
• @andre No, not really. And I could not find a similar question on mathematica.stackexchange.com either. – TMM Feb 11 '17 at 1:54
• Thanks, I asked the question only to have an idea of what is g. – andre314 Feb 11 '17 at 2:04
• Well $g$ contains some linear terms, products of the variables, logarithmic terms, trigonometric functions of the variables... There's really nothing obvious to see about $g$, so it could more-or-less be treated as a black-box function here. (Mathematica probably won't really be able to tell much from differentiating the function w.r.t. all the variables and trying to solve for $0$.) – TMM Feb 11 '17 at 2:20