# minmax¶

Contain a minmax problem solver routine.

quantecon.optimize.minmax.minmax[source]

Given an m x n matrix A, return the value $$v^*$$ of the minmax problem:

$v^* = \max_{x \in \Delta_m} \min_{y \in \Delta_n} x^T A y = \min_{y \in \Delta_n}\max_{x \in \Delta_m} x^T A y$

and the optimal solutions $$x^* \in \Delta_m$$ and $$y^* \in \Delta_n$$: $$v^* = x^{*T} A y^*$$, where $$\Delta_k = \{z \in \mathbb{R}^k_+ \mid z_1 + \cdots + z_k = 1\}$$, $$k = m, n$$.

This routine is jit-compiled by Numba, using optimize.linprog_simplex routines.

Parameters: A : ndarray(float, ndim=2) ndarray of shape (m, n). max_iter : int, optional(default=10**6) Maximum number of iteration in the linear programming solver. piv_options : PivOptions, optional PivOptions namedtuple to set tolerance values used in the linear programming solver. v : float Value $$v^*$$ of the minmax problem. x : ndarray(float, ndim=1) Optimal solution $$x^*$$, of shape (m,). y : ndarray(float, ndim=1) Optimal solution $$y^*$$, of shape (n,).