Abstract
In this paper we present a scaling algorithm for minimizing arbitrary functions over vertices of polytopes in an oracle model of computation which includes an augmentation oracle. For the binary case, when the vertices are 0–1 vectors, we show that the oracle time is polynomial. Also, this algorithm allows us to generalize some concepts of combinatorial optimization concerning performance bounds of greedy algorithms and leads to new bounds for the complexity of the simplex method.
Highlights
Where f is a function and S is a finite subset of rational vectors in Rn
In this paper we present a scaling algorithm for minimizing arbitrary functions over vertices of polytopes in an oracle model of computation which includes an augmentation oracle
We assume that f belongs to a class C, of functions defined on S, closed under additions of linear functions, i.e., if f ∈ C and h is linear over S, f + h ∈ C
Summary
Where f is a function and S is a finite subset of rational vectors in Rn. We assume that f can be accessed via an oracle which uses a black-box data structure to compute f. The scaling parameter decreases by a factor of two whenever the current solution is optimal for the perturbed objective function, which is verified by means of the augmentation oracle. Our bound can be viewed as an improvement of the bound obtained recently by Kitahara and Mizuno [7], which depends linearly on the dimension Another important consequence of our scaling algorithm is that any greedy algorithm for binary optimization, with the property that it is able to find an optimal solution for any function in the class C, must run in polynomial oracle time. This question is very nontrivial and should be studied separately
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