Complexity Theory of Real Functions pp 71-106 | Cite as

# Maximization

## Abstract

Optimization is one of the most common problems in both discrete and continuous computation. In discrete complexity theory, close connections between optimization and nondeterminism have been observed, as many combinatorial optimization problems have been shown to be *NP*-complete (cf. Garey and Johnson [1979]). On the other hand, optimization of a continuous function can often be solved in polynomial time. As an example, the linear programming problem is solvable in polynomial time [Khachiyan, 1979; Karmarkar, 1984], while the linear integer programming problem remains *NP*-hard. The question arises as to whether the continuous optimization problems are always easier than the discrete versions. In this chapter, we investigate this question on polynomial-time computable functions; more precisely, we consider the computational complexity of the maximum value of a polynomial-time computable function *f* on [0,1]. It is to be shown that these maximum values axe exactly the real numbers which have a (general) left cut in *NP* (called left *NP* real numbers). For two-dimensional, polynomial-time computable functions *f* on [0, l]^{2}, the maximum functions *g(x*) = max{*f(x*, *y*)| 0 ≤ *y* ≤ 1} coincide with *NP* real functions (real functions whose *undergraphs* are in *NP*).

## Keywords

Polynomial Time Real Function Maximum Point Computable Function Binary Expansion## Preview

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