Convex separable optimization is not much harder than linear optimization

Abstract

The polynomiality of nonlinear separable convex (concave) optimization problems, on linear constraints with a matrix with “small” subdeterminants, and the polynomiality of such integer problems, provided the inteter linear version of such problems ins polynomial, is proven. This paper presents a general-purpose algorithm for converting procedures that solves linear programming problems. The conversion is polynomial for constraint matrices with polynomially bounded subdeterminants. Among the important corollaries of the algorithm is the extension of the polynomial solvability of integer linear programming problems with totally unimodular constraint matrix, to integer-separable convex programming. An algorithm for finding a ε-accurate optimal continuous solution to the nonlinear problem that is polynomial in log(1/ε) and the input size and the largest subdeterminant of the constraint matrix is also presented. These developments are based on proximity results between the continuous and integral optimal solutions for problems with any nonlinear separable convex objective function. The practical feature of our algorithm is that is does not demand an explicit representation of the nonlinear function, only a polynomial number of function evaluations on a prespecified grid.

Optimal Preemptive Scheduling for General Target Functions

“…This target function was also studied in the past for non-preemptive scheduling [2,3,4]. Finally, in Section 4.4, we show that an algorithm due to Hochbaum and Shanthikumar [7] may be applied in order to solve the mathematical program in a polynomial time whenever the target function is separable, i.e., f (λ 1 , . .…”

“…In order to solve the corresponding mathematical program MP, we may apply the polynomial time algorithm of Hochbaum and Shanthikumar [7]. That algorithm is designed to solve minimization problems of the form…”

“…It should be noted that when f is the p -norm, p < ∞, or the threshold cost function, (70), Algorithms 4.1 and 4.4 are simpler and more efficient than the general algorithm in [7]. We need to show that our mathematical program MP falls under the framework for which that algorithm applies.…”

“…, λ m ) = m j=1 g(λ j ). It should be noted that even though the p -norm target function, (4), with p < ∞, and the threshold target function, (5), are separable, the algorithms that we offer for these cases are simpler and more efficient than the general algorithm in [7].…”

Improved randomized approximation algorithms for lot-sizing problems

“…As the objective function is convex, the relaxation (PR) can be solved in polynomial time using interior point algorithms or the algorithm by Hochbaum and Shanktikumar [9]. For systems with special structure the runnning time can be improved substantially.…”