Quantitative stability of linear infinite inequality systems under block perturbations with applications to convex systems

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l ∞(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel–Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system’s data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Cánovas et al. (SIAM J. Optim. 20, 1504–1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system’s coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.

Determination of cyclic and linear siloxanes in wastewater samples by ultrasound-assisted dispersive liquid-liquid microextraction followed by gas chromatography-mass spectrometry

A fast, simple and environmentally friendly ultrasound-assisted dispersive liquid-liquid microextraction (USA-DLLME) procedure has been developed to preconcentrate eight cyclic and linear siloxanes from wastewater samples prior to quantification by gas chromatography-mass spectrometry (GC-MS). A two-stage multivariate optimization approach has been developed employing a Plackett-Burman design for screening and selecting the significant factors involved in the USA-DLLME procedure, which was later optimized by means of a circumscribed central composite design. The optimum conditions were: extractant solvent volume, 13 µL; solvent type, chlorobenzene; sample volume, 13 mL; centrifugation speed, 2300 rpm; centrifugation time, 5 min; and sonication time, 2 min. Under the optimized experimental conditions the method gave levels of repeatability with coefficients of variation between 10 and 24% (n=7). Limits of detection were between 0.002 and 1.4 µg L−1. Calculated calibration curves gave high levels of linearity with correlation coefficient values between 0.991 and 0.9997. Finally, the proposed method was applied for the analysis of wastewater samples. Relative recovery values ranged between 71–116% showing that the matrix had a negligible effect upon extraction. To our knowledge, this is the first time that combines LLME and GC-MS for the analysis of methylsiloxanes in wastewater samples.

Robust linear semi-infinite programming duality under uncertainty

In this paper, we propose a duality theory for semi-infinite linear programming problems under uncertainty in the constraint functions, the objective function, or both, within the framework of robust optimization. We present robust duality by establishing strong duality between the robust counterpart of an uncertain semi-infinite linear program and the optimistic counterpart of its uncertain Lagrangian dual. We show that robust duality holds whenever a robust moment cone is closed and convex. We then establish that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and sufficient for robust duality. In other words, the robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. In the case of uncertain problems with affinely parameterized data uncertainty, we establish that robust duality is easily satisfied under a Slater type constraint qualification. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-infinite linear inequalities.

Robust solutions to multi-objective linear programs with uncertain data

In this paper we examine multi-objective linear programming problems in the face of data uncertainty both in the objective function and the constraints. First, we derive a formula for the radius of robust feasibility guaranteeing constraint feasibility for all possible scenarios within a specified uncertainty set under affine data parametrization. We then present numerically tractable optimality conditions for minmax robust weakly efficient solutions, i.e., the weakly efficient solutions of the robust counterpart. We also consider highly robust weakly efficient solutions, i.e., robust feasible solutions which are weakly efficient for any possible instance of the objective matrix within a specified uncertainty set, providing lower bounds for the radius of highly robust efficiency guaranteeing the existence of this type of solutions under affine and rank-1 objective data uncertainty. Finally, we provide numerically tractable optimality conditions for highly robust weakly efficient solutions.