A novel dual-decomposition method for non-convex mixed integer quadratically constrained quadratic problems

In this paper, we propose the novel p-branch-and-bound method for solving two-stage stochastic programming problems whose deterministic equivalents are represented by non-convex mixed-integer quadratically constrained quadratic programming (MIQCQP) models. The precision of the solution generated by the p-branch-and-bound method can be arbitrarily adjusted by altering the value of the precision factor p. The proposed method combines two key techniques. The first one, named p-Lagrangian decomposition, generates a mixed-integer relaxation of a dual problem with a separable structure for a primal non-convex MIQCQP problem. The second one is a version of the classical dual decomposition approach that is applied to solve the Lagrangian dual problem and ensures that integrality and non-anticipativity conditions are met in the optimal solution. The p-branch-and-bound method's efficiency has been tested on randomly generated instances and demonstrated superior performance over commercial solver Gurobi. This paper also presents a comparative analysis of the p-branch-and-bound method efficiency considering two alternative solution methods for the dual problems as a subroutine. These are the proximal bundle method and Frank-Wolfe progressive hedging. The latter algorithm relies on the interpolation of linearisation steps similar to those taken in the Frank-Wolfe method as an inner loop in the classic progressive hedging.Comment: 19 pages, 5 table

Aggregation and discretization in multistage stochastic programming

Multistage stochastic programs have applications in many areas and support policy makers in finding rational decisions that hedge against unforeseen negative events. In order to ensure computational tractability, continuous-state stochastic programs are usually discretized; and frequently, the curse of dimensionality dictates that decision stages must be aggregated. In this article we construct two discrete, stage-aggregated stochastic programs which provide upper and lower bounds on the optimal value of the original problem. The approximate problems involve finitely many decisions and constraints, thus principally allowing for numerical solutio

Solving nonlinear multicommodity flow problems by the analytic center cutting plane method

The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with Dijkstra's d-heap algorithm. An implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on well-known nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities

Generalized Decision Rule Approximations for Stochastic Programming via Liftings

Stochastic programming provides a versatile framework for decision-making under uncertainty, but the resulting optimization problems can be computationally demanding. It has recently been shown that, primal and dual linear decision rule approximations can yield tractable upper and lower bounds on the optimal value of a stochastic program. Unfortunately, linear decision rules often provide crude approximations that result in loose bounds. To address this problem, we propose a lifting technique that maps a given stochastic program to an equivalent problem on a higherdimensional probability space. We prove that solving the lifted problem in primal and dual linear decision rules provides tighter bounds than those obtained from applying linear decision rules to the original problem. We also show that there is a one-to-one correspondence between linear decision rules in the lifted problem and families of non-linear decision rules in the original problem. Finally, we identify structured liftings that give rise to highly flexible piecewise linear decision rules and assess their performance in the context of a stylized investment planning problem.

A Two-Level Approach to Large Mixed-Integer Programs with Application to Cogeneration in Energy-Efficient Buildings

We study a two-stage mixed-integer linear program (MILP) with more than 1 million binary variables in the second stage. We develop a two-level approach by constructing a semi-coarse model (coarsened with respect to variables) and a coarse model (coarsened with respect to both variables and constraints). We coarsen binary variables by selecting a small number of pre-specified daily on/off profiles. We aggregate constraints by partitioning them into groups and summing over each group. With an appropriate choice of coarsened profiles, the semi-coarse model is guaranteed to find a feasible solution of the original problem and hence provides an upper bound on the optimal solution. We show that solving a sequence of coarse models converges to the same upper bound with proven finite steps. This is achieved by adding violated constraints to coarse models until all constraints in the semi-coarse model are satisfied. We demonstrate the effectiveness of our approach in cogeneration for buildings. The coarsened models allow us to obtain good approximate solutions at a fraction of the time required by solving the original problem. Extensive numerical experiments show that the two-level approach scales to large problems that are beyond the capacity of state-of-the-art commercial MILP solvers

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

On Augmented Lagrangian Decomposition Methods for Multistage Stochastic Programs

A general decomposition framework for large convex optimization problems based on augmented Lagrangians is described. The approach is then applied to multistage stochastic programming problems in two different ways: by decomposing the problem into scenarios and by decomposing it into nodes corresponding to stages. Theoretical convergence properties of the two approaches are derived and a computational illustration is presented

Stochastic optimization of product-machine qualification in a semiconductor back-end facility

abstract: In order to process a product in a semiconductor back-end facility, a machine needs to be qualified, first by having product-specific software installed and then running test wafers through it to verify that the machine is capable of performing the process correctly. In general, not all machines are qualified to process all products due to the high machine qualification cost and tool set availability. The machine qualification decision affects future capacity allocation in the facility and subsequently affects daily production schedules. To balance the tradeoff between current machine qualification costs and future potential backorder costs due to not enough machines qualified with uncertain demand, a stochastic product–machine qualification optimization model is proposed in this article. The L-shaped method and acceleration techniques are proposed to solve the stochastic model. Computational results are provided to show the necessity of the stochastic model and the performance of different solution methods.This is an Author's Accepted Manuscript of an article published as Fu, Mengying, Askin, Ronald, Fowler, John, & Zhang, Muhong (2015). Stochastic optimization of product-machine qualification in a semiconductor back-end facility. IIE TRANSACTIONS, 47(7), 739-750. DOI: 10.1080/0740817X.2014.964887. Copyright Taylor & Francis, available online at: http://www.tandfonline.com/doi/abs/10.1080/0740817X.2014.96488