Piecewise linearisation of the first order loss function for families of arbitrarily distributed random variables

We discuss the problem of computing optimal linearisation parameters for the first order loss function of a family of arbitrarily distributed random variable. We demonstrate that, in contrast to the problem in which parameters must be determined for the loss function of a single random variable, this problem is nonlinear and features several local optima and plateaus. We introduce a simple and yet effective heuristic for determining these parameters and we demonstrate its effectiveness via a numerical analysis carried out on a well known stochastic lot sizing problem

A learning-based algorithm to quickly compute good primal solutions for Stochastic Integer Programs

We propose a novel approach using supervised learning to obtain near-optimal primal solutions for two-stage stochastic integer programming (2SIP) problems with constraints in the first and second stages. The goal of the algorithm is to predict a "representative scenario" (RS) for the problem such that, deterministically solving the 2SIP with the random realization equal to the RS, gives a near-optimal solution to the original 2SIP. Predicting an RS, instead of directly predicting a solution ensures first-stage feasibility of the solution. If the problem is known to have complete recourse, second-stage feasibility is also guaranteed. For computational testing, we learn to find an RS for a two-stage stochastic facility location problem with integer variables and linear constraints in both stages and consistently provide near-optimal solutions. Our computing times are very competitive with those of general-purpose integer programming solvers to achieve a similar solution quality

Beberapa Metode pada Masalah Pemrograman Stokastik

Stochastic programming problem is mathematical problem (linear, integer, mixed integer, and nonlinier) with stochastic element lies data. To get reasonable solution and optimal with its stochastic data is needed several method.  Applicable method in trouble stochastic programming are L-Shape decomposition and lagrange decomposition. Each method can determine optimal solution to troubleshoots stochastic programmin

A Decision Analytic Approach to Reliability-Based Design Optimization

Abstract Reliability-based design optimization is concerned with designing a product to optimize an objective function given uncertainties about whether various design constraints will be satisfied. However, the widespread practice of formulating such problems as chance-constrained programs can lead to misleading solutions. While a decision analytic approach would avoid this undesirable result, many engineers find it difficult to determine the utility functions required for a traditional decision analysis. This paper presents an alternative decision analytic formulation which, though implicitly using utility functions, is more closely related to probability maximization formulations with which engineers are comfortable and skilled. This result combines the rigor of decision analysis with the convenience of existing optimization approaches

Problem-driven scenario generation:an analytical approach for stochastic programs with tail risk measure

Scenario generation is the construction of a discrete random vector to represent parameters of uncertain values in a stochastic program. Most approaches to scenario generation are distribution-driven, that is, they attempt to construct a random vector which captures well in a probabilistic sense the uncertainty. On the other hand, a problem-driven approach may be able to exploit the structure of a problem to provide a more concise representation of the uncertainty. There have been only a few problem-driven approaches proposed, and these have been heuristic in nature. In this paper we propose what is, as far as we are aware, the first analytic approach to problem-driven scenario generation. This approach applies to stochastic programs with a tail risk measure, such as conditional value-at-risk. Since tail risk measures only depend on the upper tail of a distribution, standard methods of scenario generation, which typically spread there scenarios evenly across the support of the solution, struggle to adequately represent tail risk well

Learning-Based Matheuristic Solution Methods for Stochastic Network Design

Cette dissertation consiste en trois études, chacune constituant un article de recherche. Dans tous les trois articles, nous considérons le problème de conception de réseaux multiproduits, avec coût fixe, capacité et des demandes stochastiques en tant que programmes stochastiques en deux étapes. Dans un tel contexte, les décisions de conception sont prises dans la première étape avant que la demande réelle ne soit réalisée, tandis que les décisions de flux de la deuxième étape ajustent la solution de la première étape à la réalisation de la demande observée. Nous considérons l’incertitude de la demande comme un nombre fini de scénarios discrets, ce qui est une approche courante dans la littérature. En utilisant l’ensemble de scénarios, le problème mixte en nombre entier (MIP) résultant, appelé formulation étendue (FE), est extrêmement difficile à résoudre, sauf dans des cas triviaux. Cette thèse vise à faire progresser le corpus de connaissances en développant des algorithmes efficaces intégrant des mécanismes d’apprentissage en matheuristique, capables de traiter efficacement des problèmes stochastiques de conception pour des réseaux de grande taille. Le premier article, s’intitulé "A Learning-Based Matheuristc for Stochastic Multicommodity Network Design". Nous introduisons et décrivons formellement un nouveau mécanisme d’apprentissage basé sur l’optimisation pour extraire des informations concernant la structure de la solution du problème stochastique à partir de solutions obtenues avec des combinaisons particulières de scénarios. Nous proposons ensuite une matheuristique "Learn&Optimize", qui utilise les méthodes d’apprentissage pour déduire un ensemble de variables de conception prometteuses, en conjonction avec un solveur MIP de pointe pour résoudre un problème réduit. Le deuxième article, s’intitulé "A Reduced-Cost-Based Restriction and Refinement Matheuristic for Stochastic Network Design". Nous étudions comment concevoir efficacement des mécanismes d’apprentissage basés sur l’information duale afin de guider la détermination des variables dans le contexte de la conception de réseaux stochastiques. Ce travail examine les coûts réduits associés aux variables hors base dans les solutions déterministes pour guider la sélection des variables dans la formulation stochastique. Nous proposons plusieurs stratégies pour extraire des informations sur les coûts réduits afin de fixer un ensemble approprié de variables dans le modèle restreint. Nous proposons ensuite une approche matheuristique utilisant des techniques itératives de réduction des problèmes. Le troisième article, s’intitulé "An Integrated Learning and Progressive Hedging Method to Solve Stochastic Network Design". Ici, notre objectif principal est de concevoir une méthode de résolution capable de gérer un grand nombre de scénarios. Nous nous appuyons sur l’algorithme Progressive Hedging (PHA), ou les scénarios sont regroupés en sous-problèmes. Nous intégrons des methodes d’apprentissage au sein de PHA pour traiter une grand nombre de scénarios. Dans notre approche, les mécanismes d’apprentissage developpés dans le premier article de cette thèse sont adaptés pour résoudre les sous-problèmes multi-scénarios. Nous introduisons une nouvelle solution de référence à chaque étape d’agrégation de notre ILPH en exploitant les informations collectées à partir des sous problèmes et nous utilisons ces informations pour mettre à jour les pénalités dans PHA. Par conséquent, PHA est guidé par les informations locales fournies par la procédure d’apprentissage, résultant en une approche intégrée capable de traiter des instances complexes et de grande taille. Dans les trois articles, nous montrons, au moyen de campagnes expérimentales approfondies, l’intérêt des approches proposées en termes de temps de calcul et de qualité des solutions produites, en particulier pour traiter des cas très difficiles avec un grand nombre de scénarios.This dissertation consists of three studies, each of which constitutes a self-contained research article. In all of the three articles, we consider the multi-commodity capacitated fixed-charge network design problem with uncertain demands as a two-stage stochastic program. In such setting, design decisions are made in the first stage before the actual demand is realized, while second-stage flow-routing decisions adjust the first-stage solution to the observed demand realization. We consider the demand uncertainty as a finite number of discrete scenarios, which is a common approach in the literature. By using the scenario set, the resulting large-scale mixed integer program (MIP) problem, referred to as the extensive form (EF), is extremely hard to solve exactly in all but trivial cases. This dissertation is aimed at advancing the body of knowledge by developing efficient algorithms incorporating learning mechanisms in matheuristics, which are able to handle large scale instances of stochastic network design problems efficiently. In the first article, we propose a novel Learning-Based Matheuristic for Stochastic Network Design Problems. We introduce and formally describe a new optimizationbased learning mechanism to extract information regarding the solution structure of a stochastic problem out of the solutions of particular combinations of scenarios. We subsequently propose the Learn&Optimize matheuristic, which makes use of the learning methods in inferring a set of promising design variables, in conjunction with a state-ofthe- art MIP solver to address a reduced problem. In the second article, we introduce a Reduced-Cost-Based Restriction and Refinement Matheuristic. We study on how to efficiently design learning mechanisms based on dual information as a means of guiding variable fixing in the context of stochastic network design. The present work investigates how the reduced cost associated with non-basic variables in deterministic solutions can be leveraged to guide variable selection within stochastic formulations. We specifically propose several strategies to extract reduced cost information so as to effectively identify an appropriate set of fixed variables within a restricted model. We then propose a matheuristic approach using problem reduction techniques iteratively (i.e., defining and exploring restricted region of global solutions, as guided by applicable dual information). Finally, in the third article, our main goal is to design a solution method that is able to manage a large number of scenarios. We rely on the progressive hedging algorithm (PHA) where the scenarios are grouped in subproblems. We propose a two phase integrated learning and progressive hedging (ILPH) approach to deal with a large number of scenarios. Within our proposed approach, the learning mechanisms from the first study of this dissertation have been adapted as an efficient heuristic method to address the multi-scenario subproblems within each iteration of PHA.We introduce a new reference point within each aggregation step of our proposed ILPH by exploiting the information garnered from subproblems, and using this information to update the penalties. Consequently, the ILPH is governed and guided by the local information provided by the learning procedure, resulting in an integrated approach capable of handling very large and complex instances. In all of the three mentioned articles, we show, by means of extensive experimental campaigns, the interest of the proposed approaches in terms of computation time and solution quality, especially in dealing with very difficult instances with a large number of scenarios

Hybrid Offline/Online Methods for Optimization Under Uncertainty

This work considers multi-stage optimization problems under uncertainty. In this context, at each stage some uncertainty is revealed and some decision must be made: the need to account for multiple future developments makes stochastic optimization incredibly challenging. Due to such a complexity, the most popular approaches depend on the temporal granularity of the decisions to be made. These approaches are, in general, sampling-based methods and heuristics. Long-term strategic decisions (which are often very impactful) are typically solved via expensive, but more accurate, sampling-based approaches. Short-term operational decisions often need to be made over multiple steps, within a short time frame: they are commonly addressed via polynomial-time heuristics, while more advanced sampling-based methods are applicable only if their computational cost is carefully managed. We will refer to the first class of problems (and solution approaches) as offline and to the second as online. These phases are typically solved in isolation, despite being strongly interconnected. Starting from the idea of providing multiple options to balance the solution quality/time trade-off in optimization problem featuring offline and online phases, we propose different methods that have broad applicability. These methods have been firstly motivated by applications in real-word energy problems that involve distinct offline and online phases: for example, in Distributed Energy Management Systems we may need to define (offline) a daily production schedule for an industrial plant, and then manage (online) its power supply on a hour by hour basis. Then we show that our methods can be applied to a variety of practical application scenarios in very different domains with both discrete and numeric decision variables

Piecewise linear approximations of the standard normal first order loss function

The first order loss function and its complementary function are extensively used in practical settings. When the random variable of interest is normally distributed, the first order loss function can be easily expressed in terms of the standard normal cumulative distribution and probability density function. However, the standard normal cumulative distribution does not admit a closed form solution and cannot be easily linearised. Several works in the literature discuss approximations for either the standard normal cumulative distribution or the first order loss function and their inverse. However, a comprehensive study on piecewise linear upper and lower bounds for the first order loss function is still missing. In this work, we initially summarise a number of distribution independent results for the first order loss function and its complementary function. We then extend this discussion by focusing first on random variable featuring a symmetric distribution, and then on normally distributed random variables. For the latter, we develop effective piecewise linear upper and lower bounds that can be immediately embedded in MILP models. These linearisations rely on constant parameters that are independent of the mean and standard deviation of the normal distribution of interest. We finally discuss how to compute optimal linearisation parameters that minimise the maximum approximation error.Comment: 22 pages, 7 figures, working draf