Substitutes and complements in network flows viewed as discrete convexity
Discrete Optimization
Ordered optimal solutions and parametric minimum cut problems
Discrete Optimization
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Optimal selections are parameter-dependent optimal solutions of parametric optimization problems whose properties can be used in sensitivity analysis. Here we present a qualitative theory of sensitivity analysis for linearly-constrained convex separable (i.e., monotropic) parametric optimization problems. Three qualitative sensitivity analysis results previously derived for network flows are extended to monotropic problems: The Ripple and Smoothing Theorems give upper bounds on the magnitude of optimal-variable variations as a function of variations in the problem parameter(s), the theory of substitutes and complements provides necessary and sufficient conditions for optimal-variable changes to consistently have the same (or the opposite) sign(s) in two given variables, and the Monotonicity Theorem links changes in the value of the parameters to changes in optimal decision variables. We introduce a class of optimal selections for which these results hold, thereby answering a long-standing question due to Granot and Veinott (1985) with a simple and elegant method. Although a number of results are known to depend on the resolution of NP-complete problems, easily computable nonnetwork classes of monotropic problems such as unimodular systems of constraints emerge in the light of the present approach.