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- ItemOn convex lower-level black-box constraints in bilevel optimization with an application to gas market models with chance constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Heitsch, Holger; Henrion, René; Kleinert, Thomas; Schmidt, MartinBilevel optimization is an increasingly important tool to model hierarchical decision making. However, the ability of modeling such settings makes bilevel problems hard to solve in theory and practice. In this paper, we add on the general difficulty of this class of problems by further incorporating convex black-box constraints in the lower level. For this setup, we develop a cutting-plane algorithm that computes approximate bilevel-feasible points. We apply this method to a bilevel model of the European gas market in which we use a joint chance constraint to model uncertain loads. Since the chance constraint is not available in closed form, this fits into the black-box setting studied before. For the applied model, we use further problem-specific insights to derive bounds on the objective value of the bilevel problem. By doing so, we are able to show that we solve the application problem to approximate global optimality. In our numerical case study we are thus able to evaluate the welfare sensitivity in dependence of the achieved safety level of uncertain load coverage.
- ItemOn M-stationarity conditions in MPECs and the associated qualification conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Adam, Lukáš; Henrion, René; Outrata, Jir̆íDepending on whether a mathematical program with equilibrium constraints (MPEC) is considered in its original or its enhanced (via KKT conditions) form, the assumed constraint qualifications (CQs) as well as the derived necessary optimality conditions may differ significantly. In this paper, we study this issue when imposing one of the weakest possible CQs, namely the calmness of the perturbation mapping associated with the respective generalized equations in both forms of the MPEC. It is well known that the calmness property allows one to derive socalled M-stationarity conditions. The strength of assumptions and conclusions in the two forms of the MPEC is strongly related with the CQs on the lower level imposed on the set whose normal cone appears in the generalized equation. For instance, under just the Mangasarian-Fromovitz CQ (a minimum assumption required for this set), the calmness properties of the original and the enhanced perturbation mapping are drastically different. They become identical in the case of a polyhedral set or when adding the Full Rank CQ. On the other hand, the resulting optimality conditions are affected too. If the considered set even satisfies the Linear Independence CQ, both the calmness assumption and the derived optimality conditions are fully equivalent for the original and the enhanced form of the MPEC. A compilation of practically relevant consequences of our analysis in the derivation of necessary optimality conditions is provided in the main Theorem 4.3. The obtained results are finally applied to MPECs with structured equilibria.
- ItemDynamic probabilistic constraints under continuous random distributions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) González Grandón, Tatiana; Henrion, René; Pérez-Aros, PedroThe paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi-) continuity of the probability function or existence of solutions are studied. It turns out that the results differ significantly according to whether decision rules are embedded into Lebesgue or Sobolev spaces. In the second part, the simplest meaningful two-stage model with decision rules from L 2 is investigated. More specific properties like Lipschitz continuity and differentiability of the probability function are considered. Explicitly verifiable conditions for these properties are provided along with explicit gradient formulae in the Gaussian case. The application of such formulae in the context of necessary optimality conditions is discussed and a concrete identification of solutions presented.
- ItemStability and sensitivity of optimization problems with first order stochastic dominance constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Dentcheva, Darinka; Henrion, René; Ruszczynski, AndrzejWe analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order. We consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance. We show that the graph of the feasible set mapping is closed under rather general assumptions. We obtain conditions for the continuity of the optimal value and upper-semicontinuity of the optimal solutions, as well as quantitative stability estimates of Lipschitz type. Furthermore, we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional derivatives of the optimal value. The estimates are formulated in terms of the dual utility functions associated with the dominance constraints.
- ItemConditioning of linear-quadratic two-stage stochastic optimization problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Emich, Konstantin; Henrion, René; Römisch, WernerIn this paper a condition number for linear-quadratic two-stage stochastic optimization problems is introduced as the Lipschitz modulus of the multifunction assigning to a (discrete) probability distribution the solution set of the problem. Being the outer norm of the Mordukhovich coderivative of this multifunction, the condition number can be estimated from above explicitly in terms of the problem data by applying appropriate calculus rules. Here, a chain rule for the extended partial second-order subdifferential recently proved by Mordukhovich and Rockafellar plays a crucial role. The obtained results are illustrated for the example of two-stage stochastic optimization problems with simple recourse.
- ItemA simple formula for the second-order subdifferential of maximum functions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Emich, Konstantin; Henrion, RenéWe derive a simple formula for the second-order subdifferential of the maximum of coordinates which allows us to construct this set immediately from its argument and the direction to which it is applied. This formula can be combined with a chain rule recently proved by Mordukhovich and Rockafellar [9] in order to derive a similarly simple formula for the extended partial second-order subdifferential of finite maxima of smooth functions. Analogous formulae can be derived immediately for the full and conventional partial secondorder subdifferentials.
- ItemOn calculating the normal cone to a finite union of convex polyhedra(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Henrion, René; Outrata, JiříThe paper provides formulae for calculating the limiting normal cone introduced by Mordukhovich to a finite union of convex polyhedra. In the first part, special cases of independent interest are considered (almost disjoint cones, half spaces, orthants). The second part focusses on unions of general polyhedra. Due to the local nature of the normal cone, one may restrict considerations without loss of generality to finite unions of polyhedral cones. First, an explicit formula for the normal cone is provided in the situation of two cones. An algorithmic approach is presented along with a refined, more efficient formula. Afterwards, a general formula for the union of N cones is derived. Finally, an application to the stability analysis of a special type of probabilistic constraints is provided.
- ItemJoint model of probabilistic-robust (probust) constraints with application to gas network optimization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Adelhütte, Dennis; Aßmann, Denis; Grandón, Tatiana González; Gugat, Martin; Heitsch, Holger; Henrion, René; Liers, Frauke; Nitsche, Sabrina; Schultz, Rüdiger; Stingl, Michael; Wintergerst, DavidOptimization problems under uncertain conditions abound in many real-life applications. While solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we introduce a class of joint probabilistic/robust constraints. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for finding their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound with high probability.
- ItemA turnpike property for optimal control problems with dynamic probabilistic constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Gugat, Martin; Heitsch, Holger; Henrion, RenéIn this paper we consider systems that are governed by linear time-discrete dynamics with an initial condition, additive random perturbations in each step and a terminal condition for the expected values. We study optimal control problems where the objective function consists of a term of tracking type for the expected values and a control cost. In addition, the feasible states have to satisfy a conservative probabilistic constraint that requires that the probability that the trajectories remain in a given set F is greater than or equal to a given lower bound. An application are optimal control problems related to storage management systems with uncertain in- and output. We give sufficient conditions that imply that the optimal expected trajectories remain close to a certain state that can be characterized as the solution of an optimal control problem without prescribed initial- and terminal condition. In this way we contribute to the study of the turnpike phenomenon that is well-known in mathematical economics and make a step towards the extension of the turnpike theory to problems with probabilistic constraints.
- ItemControlled polyhedral sweeping processes: Existence, stability, and optimality conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Henrion, René; Jourani, Abderrahim; Mordukhovich, Boris S.This paper is mainly devoted to the study of controlled sweeping processes with polyhedral moving sets in Hilbert spaces. Based on a detailed analysis of truncated Hausdorff distances between moving polyhedra, we derive new existence and uniqueness theorems for sweeping trajectories corresponding to various classes of control functions acting in moving sets. Then we establish quantitative stability results, which provide efficient estimates on the sweeping trajectory dependence on controls and initial values. Our final topic, accomplished in finite-dimensional state spaces, is deriving new necessary optimality and suboptimality conditions for sweeping control systems with endpoint constrains by using constructive discrete approximations.