Complexity Theory of Real Functions pp 274-289 | Cite as

# An Optimization Problem in Control Theory

Chapter

## Abstract

The following optimization problem has arisen from the non-classical control theory: for a given continuous function Intuitively, the function

*f*: [0, 1]^{4}→ [0, 1] that satisfies the Lipschitz condition, compute the minimum value$${{\alpha }^{*}}(f) = \mathop{{\inf }}\limits_{{{{g}_{1}},{{g}_{2}} \in C[0,1]}} \int_{0}^{1} {\int_{0}^{1} {f({{x}_{1}},{{x}_{2}},{{g}_{1}}({{x}_{1}}),g({{x}_{2}}))d{{x}_{1}}d{{x}_{2}}.} }$$

(9.1)

*f*may be viewed as the cost function on inputs*x*_{1},*x*_{2}∈ [0, 1] and the corresponding decisions*g*_{1}(*x*) and*g*_{2}(*x*_{2}) on these inputs. The decision functions*g*_{1}and*g*_{2}are based only on part of the input values and perform, in a sense, in the distributed manner. The goal is to find the best distributed decision functions that minimize the average cost*f*. The condition that*f*satisfies the Lipschitz condition is necessary so that the minimum value*α**(*f*) is computable. For more about the motivations and applications of this problem, see Papadimitriou and Tsitsiklis [1982, 1986].## Keywords

Lipschitz Condition Boolean Variable Exponential Time Truth Assignment Follow Optimization Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Birkhäuser Boston 1991