Abstract
We present a novel technique for obtaining global solutions to discrete min-max problems that arise naturally in the receding horizon control of unmanned craft in which the controls can be adjusted only in notches, e.g., stop, half forward, full forward, left or right \(60^{\circ }\), and in the finite precision global solution of certain classes of semi-infinite min-max problems. The technique consists of a method for transcribing min-max problems over discrete sets into a matrix game and matrix game-specific adaptations of the Method of Outer Approximations and the Method of Adaptive Approximations, which are normally used for solving optimal control and semi-infinite min-max problems. The efficiency of the Method of Outer Approximations depends on having a good initial approximation to a solution. To this end, we make use of adaptive approximation techniques to decompose a large matrix game into a sequence of lower dimensional games, the solution of each giving rise to a very good initial approximation to a solution for the next game. We show that a basic approach for solving a min-max matrix game, in which one maximizes over the elements of columns and minimizes over the elements of the rows of a matrix, requires a number of function evaluations which is equal to the product of the number of rows and the number of columns of the matrix, while with our new approach our experimental results require only a number of function evaluations which is the product of the number rows and a number ranging from 2 to 20, shortening computing times from years to fractions of a minute.
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Notes
As stated, this method is conceptual because it requires global optimization evaluations.
In practice, it may be helpful to augment the set \(Y_0\) with a few more well-chosen points.
We say that a sequence of problems is nested if a feasible solution for the \(i-th\) problem is also a feasible solution for the \((i+1)-th\) problem.
When the discretizations are finite, as in the matrix games that we consider later, the termination test is trivial: the problem is solved to completion in finite time.
A more realistic statement of the receding horizon control algorithm includes provisions for the time it takes to solve the appropriate min-max problem of the form (12).
The cardinality of \(\mathbf I\) and \(\mathbf J\) is obviously k.
The number of points in X and Y, respectively, is \(N+1\).
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Communicated by David Q. Mayne.
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Polak, E., Lee, S., Bustany, I. et al. Method of Outer Approximations and Adaptive Approximations for a Class of Matrix Games. J Optim Theory Appl 170, 876–899 (2016). https://doi.org/10.1007/s10957-016-0953-7
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DOI: https://doi.org/10.1007/s10957-016-0953-7
Keywords
- Method of Outer Approximations
- Adaptive discretization
- Discrete min-max problems
- Receding horizon control
- Global optimization