05031 Abstracts Collection -- Algorithms for Optimization with Incomplete Information

From 16.01.05 to 21.01.05, the Dagstuhl Seminar 05031 ``Algorithms for Optimization with Incomplete Information\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

On the relationship between bilevel decomposition algorithms and direct interior-point methods

Engineers have been using bilevel decomposition algorithms to solve certain nonconvex large-scale optimization problems arising in engineering design projects. These algorithms transform the large-scale problem into a bilevel program with one upperlevel problem (the master problem) and several lower-level problems (the subproblems). Unfortunately, there is analytical and numerical evidence that some of these commonly used bilevel decomposition algorithms may fail to converge even when the starting point is very close to the minimizer. In this paper, we establish a relationship between a particular bilevel decomposition algorithm, which only performs one iteration of an interior-point method when solving the subproblems, and a direct interior-point method, which solves the problem in its original (integrated) form. Using this relationship, we formally prove that the bilevel decomposition algorithm converges locally at a superlinear rate. The relevance of our analysis is that it bridges the gap between the incipient local convergence theory of bilevel decomposition algorithms and the mature theory of direct interior-point methods

On the Value of Multistage Risk-Averse Stochastic Facility Location With or Without Prioritization

We consider a multiperiod stochastic capacitated facility location problem under uncertain demand and budget in each period. Using a scenario tree representation of the uncertainties, we formulate a multistage stochastic integer program to dynamically locate facilities in each period and compare it with a two-stage approach that determines the facility locations up front. In the multistage model, in each stage, a decision maker optimizes facility locations and recourse flows from open facilities to demand sites, to minimize certain risk measures of the cost associated with current facility location and shipment decisions. When the budget is also uncertain, a popular modeling framework is to prioritize the candidate sites. In the two-stage model, the priority list is decided in advance and fixed through all periods, while in the multistage model, the priority list can change adaptively. In each period, the decision maker follows the priority list to open facilities according to the realized budget, and optimizes recourse flows given the realized demand. Using expected conditional risk measures (ECRMs), we derive tight lower bounds for the gaps between the optimal objective values of risk-averse multistage models and their two-stage counterparts in both settings with and without prioritization. Moreover, we propose two approximation algorithms to efficiently solve risk-averse two-stage and multistage models without prioritization, which are asymptotically optimal under an expanding market assumption. We also design a set of super-valid inequalities for risk-averse two-stage and multistage stochastic programs with prioritization to reduce the computational time. We conduct numerical studies using both randomly generated and real-world instances with diverse sizes, to demonstrate the tightness of the analytical bounds and efficacy of the approximation algorithms and prioritization cuts

Some experiments on solving multistage stochastic mixed 0-1 programs with time stochastic dominance constraints

In this work we extend to the multistage case two recent risk averse measures for two-stage stochastic programs based on first- and second-order stochastic dominance constraints induced by mixed-integer linear recourse. Additionally, we consider Time Stochastic Dominance (TSD) along a given horizon. Given the dimensions of medium-sized problems augmented by the new variables and constraints required by those risk measures, it is unrealistic to solve the problem up to optimality by plain use of MIP solvers in a reasonable computing time, at least. Instead of it, decomposition algorithms of some type should be used. We present an extension of our Branch-and-Fix Coordination algorithm, so named BFC-TSD, where a special treatment is given to cross scenario group constraints that link variables from different scenario groups. A broad computational experience is presented by comparing the risk neutral approach and the tested risk averse strategies. The performance of the new version of the BFC algorithm versus the plain use of a state-of-the-artMIP solver is also reported

A time-consistent Benders decomposition method for multistage distributionally robust stochastic optimization with a scenario tree structure

A computational method is developed for solving time consistent distributionally robust multistage stochastic linear programs with discrete distribution. The stochastic structure of the uncertain parameters is described by a scenario tree. At each node of this tree, an ambiguity set is defined by conditional moment constraints to guarantee time consistency. This method employs the idea of nested Benders decomposition that incorporates forward and backward steps. The backward steps solve some conic programming problems to approximate the cost-to-go function at each node, while the forward steps are used to generate additional trial points. A new framework of convergence analysis is developed to establish the global convergence of the approximation procedure, which does not depend on the assumption of polyhedral structure of the original problem. Numerical results of a practical inventory model are reported to demonstrate the effectiveness of the proposed method

Power management in a hydro-thermal system under uncertainty by Lagrangian relaxation

We present a dynamic multistage stochastic programming model for the cost-optimal generation of electric power in a hydro-thermal system under uncertainty in load, inflow to reservoirs and prices for fuel and delivery contracts. The stochastic load process is approximated by a scenario tree obtained by adapting a SARIMA model to historical data, using empirical means and variances of simulated scenarios to construct an initial tree, and reducing it by a scenario deletion procedure based on a suitable probability distance. Our model involves many mixed-integer variables and individual power unit constraints, but relatively few coupling constraints. Hence we employ stochastic Lagrangian relaxation that assigns stochastic multipliers to the coupling constraints. Solving the Lagarangian dual by a proximal bundle method leads to successive decomposition into single thermal and hydro unit subproblems that are solved by dynamic programming and a specialized descent algorithm, respectively. The optimal stochastic multipliers are used in Lagrangian heuristics to construct approximately optimal first stage decisions. Numerical results are presented for realistic data from a German power utility, with a time horizon of one week and scenario numbers ranging from 5 to 100. The corresponding optimization problems have up to 200,000 binary and 350,000 continuous variables, and more than 500,000 constraints

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

A Gradient Descent-Ascent Method for Continuous-Time Risk-Averse Optimal Control

In this paper, we consider continuous-time stochastic optimal control problems where the cost is evaluated through a coherent risk measure. We provide an explicit gradient descent-ascent algorithm which applies to problems subject to non-linear stochastic differential equations. More specifically, we leverage duality properties of coherent risk measures to relax the problem via a smooth min-max reformulation which induces artificial strong concavity in the max subproblem. We then formulate necessary conditions of optimality for this relaxed problem which we leverage to prove convergence of the gradient descent-ascent algorithm to candidate solutions of the original problem. Finally, we showcase the efficiency of our algorithm through numerical simulations involving trajectory tracking problems and highlight the benefit of favoring risk measures over classical expectation

On Multistage Stochastic and Distributionally Robust Optimization: New Algorithms, Complexity Analysis, and Performance Comparison

Multistage optimization under uncertainty refers to sequential decision-making with the presence of uncertainty information that is revealed partially until the end of planning horizon. Depending on the uncertainty model, it is often studied as multistage stochastic optimization (MSO), where one seeks optimal decisions with minimum mean objective with respect to a certain probabilistic uncertainty model; or more generally multistage distributionally robust optimization (MDRO), where one seeks optimal decisions with respect to a worst-case probability distribution over a candidate set of distributions. Both approaches have found ubiquitous applications such as in energy system and inventory planning. First, we focus on MSO with possibly integer variables and nonlinear constraints. We develop dual dynamic programming (DDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. Several interesting classes of MSO problems are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. We also characterize the iteration complexity of the proposed algorithms, which reveals that the iteration complexity depends polynomially on the number of stages. We further show that the iteration complexity depends linearly on the number of stages $T$, if all the state spaces are finite sets, or if we allow the optimality gap to scale with $T$. This complexity study resolves an open question on the iteration complexity of DDP-type algorithms. Second, we propose a new class of algorithms for solving convex MDRO problems, namely a consecutive dual dynamic programming (DDP) algorithm and a nonconsecutive version. The new algorithms generalize and strengthen existing DDP-type algorithms by introducing an important technique of regularization that enables the algorithms to handle much broader classes of MDRO problems. We then define single stage subproblem oracles (SSSO) and provide a thorough complexity analysis of the new algorithms, proving both upper complexity bounds and a matching lower bound. Numerical examples on inventory problems and hydrothermal power system planning problems are given to show the effectiveness of the proposed regularization technique. Third, we consider convex MDRO with Wasserstein ambiguity sets constructed from stagewise independent empirical distributions. We show that these data-driven MDRO models have favorable out-of-sample performance guarantees and adjustable levels of in-sample conservatism. Then we extend the DDP algorithms to the data-driven MDRO by proposing two SSSO realizations that are able to handle the Wasserstein ambiguity sets, exploiting either convexity or concavity of the uncertain cost functions, which happens when the uncertainty only appears in the right-hand-side of the constraints or in the objective function. Extensive numerical experiments on inventory problems are then conducted to compare these data-driven MDRO models with the widely used risk-neutral and risk-averse empirical MSO models.Ph.D