On accuracy of solutions for discrete optimization problems with perturbed coefficientsof the objective function

Abstract

An approach to the sensitivity analysis for discrete optimization problems with perturbedobjective function is presented. The problem is stated in the following general form:minΣ_e∈Y c(e) : Y ∈ F, where c = (c(e), e ∈ E) is a vector of weights for some finite set E,and F ⫅ 2^E is a given family of feasible subsets. It is assumed that the set of feasiblesolutions F is fixed, but the coefficients of the vector c may vary. The main problem consideredin this paper concerns the following question: How does the maximum relative errorof a given feasible solution depend on inaccuracies of the problem data? Two particularperturbations of the vector c are considered:(i) it is assumed that c is in a closed ball with radius δ≥0 and center c ^o≥0 or(ii) the relative deviation of c(e) from the value c^o(e) is not greater than δ for any e ∈ E, i.e.,|c(e) ‐ c^o(e)| ≤δ c^o(e), e ∈ E.For a given feasible solution X ∈ F , two functions of the parameter δ are introduced: thesensitivity function s(X, δ) and the accuracy function a(X, δ). The values of s(X, δ) anda(X, δ) are equal to the maximum relative error of the solution X when the perturbations ofproblem data are of the type (i) or (ii), respectively. Some general, but computationallyinefficient formulae for calculating these functions are derived and their properties arestudied. It is shown that s(X, δ) and a(X, δ) are nondecreasing, convex functions with alimited number of breakpoints. A practically efficient method of calculating upper and lowerenvelopes for the accuracy function is presented. This method is based on the notion of k‐bestsolutions of the problem. It gives an interval to which the maximum relative error ofthe solution X must belong when the coefficients of vector c are given with accuracy δ. Theapproach is illustrated with an example of the symmetric traveling salesman problem.