Quantitative Stability of Variational Systems: I. The Epigraphical Distance

This paper proposes a global measure for the distance between the elements of a variational system (parametrized families of optimization problems)

On calmness of the optimal value function

The paper is devoted to the calmness from below/from above for the optimal value function of parametric optimization problems, where we are specifically interested in perturbed semi-infinite programs. A main intention is to revisit classical results and to derive refinements of them. In particular, we show in the context of semi-infinite optimization that calmness from below for j holds under quasiconvexity of the data functions and compactness of the solution set, which extends results on the lower semicontinuity of Illustrative examples are given, which demonstrate the significance of the imposed assumptions even in the case of linear and quadratic programs

Calmness of the Optimal Value in Linear Programming

This research has been partially supported by grant MTM2014-59179-C2-2-P from MINECO, Spain, and FEDER "Una manera de hacer Europa," European Union

Explicit feedback synthesis for nonlinear robust model predictive control driven by quasi-interpolation

We present QuIFS (Quasi-Interpolation driven Feedback Synthesis): an offline feedback synthesis algorithm for explicit nonlinear robust minmax model predictive control (MPC) problems with guaranteed quality of approximation. The underlying technique is driven by a particular type of grid-based quasi-interpolation scheme. The QuIFS algorithm departs drastically from conventional approximation algorithms that are employed in the MPC industry (in particular, it is neither based on multi-parametric programming tools and nor does it involve kernel methods), and the essence of its point of departure is encoded in the following challenge-answer approach: Given an error margin $\varepsilon>0$, compute in a single stroke a feasible feedback policy that is uniformly $\varepsilon$-close to the optimal MPC feedback policy for a given nonlinear system subjected to constraints and bounded uncertainties. Closed-loop stability and recursive feasibility under the approximate feedback policy are also established. We provide a library of numerical examples to illustrate our results.Comment: 31 Page

Worst-case estimation for econometric models with unobservable components

A worst-case estimator for econometric models containing unobservable components, based on minimax principles for optimal selection of parameters, is proposed. Worst-case estimators are robust against the averse effects of unobservables. Computing worstcase estimators involves solving a minimax continuous problem, which is quite a challenging task. Large sample theory is considered, and a Monte Carlo study of finite-sample properties is conducted. A financial application is consideredPublicad

Convergence analysis under consistent error bounds

We introduce the notion of consistent error bound functions which provides a unifying framework for error bounds for multiple convex sets. This framework goes beyond the classical Lipschitzian and H\"olderian error bounds and includes logarithmic and entropic error bounds found in the exponential cone. It also includes the error bounds obtainable under the theory of amenable cones. Our main result is that the convergence rate of several projection algorithms for feasibility problems can be expressed explicitly in terms of the underlying consistent error bound function. Another feature is the usage of Karamata theory and functions of regular variations which allows us to reason about convergence rates while bypassing certain complicated expressions. Finally, applications to conic feasibility problems are given and we show that a number of algorithms have convergence rates depending explicitly on the singularity degree of the problem.Comment: 43 pages, comments welcome. To appear in Foundations of Computational Mathematic

Necessary and Sufficient Optimality Conditions in DC Semi-infinite Programming

This paper deals with particular families of DC optimization problems involving suprema of convex functions. We show that the specific structure of this type of function allows us to cover a variety of problems in nonconvex programming. Necessary and sufficient optimality conditions for these families of DC optimization problems are established, where some of these structural features are conveniently exploited. More precisely, we derive necessary and sufficient conditions for (global and local) optimality in DC semi-infinite programming and DC cone-constrained optimization, under natural constraint qualifications. Finally, a penalty approach to DC abstract programming problems is developed in the last section.The first author was partially supported by ANID Chile under grant Fondecyt Regular 1190110. The second author is supported by Research Project PGC2018-097960-B-C21 from MICINN Spain, and Australian ARC–Discovery Projects DP 180100602. The third author was partially supported by ANID Chile under grants Fondecyt regular 1190110, Fondecyt regular 1200283, and Programa Regional Mathamsud 20-Math-08 CODE: MATH190003

A complementarity approach to multistage stochastic linear programs

Das Gebiet der Stochastischen Programmierung gehört in die Problemklasse der "Entscheidungsfindung unter Unsicherheit". Anwendungen finden sich weitverbreitet in den Feldern der industriellen Produktion und der finanziellen Planung neben vielen anderen. Die Arbeit befasst sich mit der Approximation von 'Multistage Stochastic Linear Programs' (MSLP), wo einige Modelldaten als zufällig vorausgesetzt werden und sich sukzessiv in diskreter Zeit $t=1,...,T$ realisieren, wobei $T$ ein endlicher Planungshorizont sei. Entscheidungen zum Zeitpunkt $t$ sollen so gefällt werden, dass die Summe ihrer unmittelbar anfallenden Kosten und den erwarteten Recourse Kosten minimiert wird, gegeben die vorangegangenen Entscheidungen und die Information, welche bis $t$ verfügbar ist. Falls die Anzahl Szenarien endlich ist, dann lässt sich das Optimierungsproblem als Linearprogramm formulieren und auch direkt lösen, sofern diese Anzahl nicht zu gross ist. Numerische Approximationsmethoden sind häufig unumgänglich, insbesondere falls die zufälligen Daten stetig verteilt sind. Es gibt einige Methoden für den Fall $T=2$, welche auf diese Situation zugeschnitten sind. Leider stellten sich diese als unpraktisch heraus, um sie auf den Fall $T≥3$ zu erweitern, weil in diesem Fall die Auswertung eines einzelnen Recourse Funktionswertes nahezu denselben Schwierigkeitsgrad wie die Bestimmung des optimalen Zielfunktionswertes des Gesamtproblems aufweist. Da wir den Fall von stetig verteilten Daten miteinschliessen, wird MSLP als infinites Linearprogramm formuliert, welches auch eine infinite duale Form besitzt. Die Optimalitätslücke eines zulässigen primal-dual Paares kann als Erwartungswert einer nichtnegativen Zufallsvariablen ausgedrückt werden, in der Arbeit 'Komplementaritätsvariable' genannt. Eine Aggregation von Restriktionen und Entscheidungen scheint ein natürlicher Zugangzu sein, um MSLP numerisch handhabbar zu machen. Wir analysieren vor allem Modelle, bei denen jede optimale Lösung eines geeignet aggregierten Dualproblems zulässig im originalen Dualproblem ist, was auf untere Schranken führt. Danach schlagen wir einen Weg basierend auf den aggregierten Lösungen vor, wie sich rekursiv durch das Lösen einer Folge von kleinen linearen und quadratischen Subproblemen eine zulässige Entscheidungspolitik in der Originalaufgabe definieren lässt. Unter geeigneten Modellannahmen und abhängig vom Aggregationsfehler erweist sich diese Entscheidungspolitik als nahe liegend zu der aggregierten optimalen Primallösung. Ausserdem wird das Worst-Case Verhalten der Komplementaritätsvariable, welche sich aus der rekursiven Entscheidungspolitik und der aggregierten optimalen Duallösung ergibt, sowohl in Erwartung als auch in einem fast sicheren Sinn analysiert. Das letztere Resultat wird verwendet, um die Endlichkeit des vorgeschlagenen Verfeinerungsalgorithmus MSLP-APPROX nachzuweisen, welcher auf simulierten Werten der Komplementaritätsvariable basiert. Wir beweisen auch, dass - bei sukzessiver Erhöhung der Stichprobe und eines Genauigkeitsparameters von MSLP-APPROX - die (schwachen) Häufungspunkte der Lösungskandidaten das Originalproblem lösen. Um das praktische Verhalten von MSLP-APPROX zu veranschaulichen, werden im letzten Teil numerische Resultate präsentiert. The field of Stochastic Programming belongs to the problem class of "Decision-Making under Uncertainty''. Applications are widely available in the areas of industrial production and financial planning, among many others. The thesis deals with the approximation of Multistage Stochastic Linear Programs (MSLP) where some model data are assumed to be random and successively realized at time $t=1,...,T$ where $T$ is a finite planning horizon. Decisions at time $t$ should be made such that the sum of their immediate costs and the expected recourse costs is minimized, given the previous decisions and the information available up to $t$. When the number of scenarios is finite, the optimization problem can be formulated as a linear program and may also be solved directly, provided that this number is not too high. Numerical approximation methods are often inevitable, especially if the random data are continuously distributed. There are some methods available for the case $T=2$ designed for this situation. Unfortunately, they turned out to be impractical to extend to the case $T≥3$ because, in this case, the computation of a single recourse function value has almost the same degree of difficulty as determining the optimal objective value of the overall problem.Since we include the case of continuously distributed data, MSLP is expressed as an infinite linear program which also has an infinite dual form. The optimality gap of a feasible primal-dual pair is expressed as the expectation of a nonnegative random variable, in the thesis called the 'complementarity variable'. Aggregation of constraints and decisions seems to be a natural approach to make MSLP numerically manageable. We analyze particularly models where every optimal solution of a suitably aggregated dual problem is feasible in the original dual problem, leading to lower bounds. After that, based on the aggregated solutions, we propose a way to define recursively a feasible decision policy in the original primal problem by solving a sequence of small linear and quadratic subproblems. Under suitable model assumptions and depending on the aggregation error, the recursive decision policy turns out to be close to the aggregated optimal primal solution. Furthermore, the worst-case behavior of the complementarity variable resulting from the recursive decision policy and the aggregated optimal dual solution is analyzed both in expectation and in an almost sure sense. The latter result is used to prove the finiteness of the proposed refinement algorithm MSLP-APPROX which is based on simulated values of the complementarity variable. We also prove that - by successively increasing both the sample size and an accuracy parameter of MSLP-APPROX} - the (weak) accumulation points of the candidate solutions solve the original problem. In the last part, numerical results are presented in order to illustrate the practical behavior of MSLP-APPROX