Differentiable selection of optimal solutions in parametric linear programming

Abstract

In the present paper we prove that if the data of a parametric linear optimization problem are smooth, the solution map admits a local smooth selection "almost" everywhere. This in particular shows that the set of points where the marginal function of the problem is nondifferentiable is nowhere dense.

The Optimal Power Flow Operator: Theory and Computation

“…Results such as those above have been shown to hold (with high probability) when the problem is simply viewed as a mathematical program (see for example; [9], [10], [11], [12]), of which the DC-OPF problem is a special case. However these results, while insightful in optimization theory convey little information about the process the optimization problem models.…”

“…Proof: By checking the conditions 1-4 in Lemma 1, the proof is established. Alternatively, the theorem can be proved by extending Proposition 3.2 in [11].…”

Local smooth representation of solution sets in parametric linear fractional programming problems

“…According to the Theorem 3.1, the optimal solution set of this problem can be represented as follows: S(ω) = x ∈ R n : x = v 1 (ω) + µd 1 (ω), µ ≥ 0 . [8], Luc and Dien [9] proved that a parametric polyhedron with smooth data has a local smooth representation. As a consequence, they proposed the smooth representation technique to investigate the sensitivity of the linear programming problem by proving that the solution set of a parametric linear programming problem with smooth data has a local smooth representation.…”

“…Sensitivity analysis [2,7] involves determining how much changes in the parametric optimization problems influence the optimal solution sets where the optimal value can be attained. By using the smooth representation technique for the parametric polyhedra, Luc [8], Luc and Dien [9] proved that a parametric linear programming problem with smooth data has a polyhedra optimal solution set which has a local smooth representation. Fang [5] further developed the smooth representation technique for parametric semiclosed polyhedra and applied it to prove that the solution set of a smooth parametric piecewise linear programming problems can be locally represented as a finite union of parametric semiclosed polyhedra generated by finite smooth functions.…”