Decomposition-Based Method for Sparse Semidefinite Relaxations of Polynomial Optimization Problems

We consider polynomial optimization problems pervaded by a sparsity pattern. It has been shown in [1, 2] that the optimal solution of a polynomial programming problem with structured sparsity can be computed by solving a series of semidefinite relaxations that possess the same kind of sparsity. We aim at solving the former relaxations with a decompositionbased method, which partitions the relaxations according to their sparsity pattern. The decomposition-based method that we propose is an extension to semidefinite programming of the Benders decomposition for linear programs [3] .Polynomial optimization, Semidefinite programming, Sparse SDP relaxations, Benders decomposition

An interior-point and decomposition approach to multiple stage stochastic programming

There is no abstract of this report

An interior-point and decomposition approach to multiple stage stochastic programming

There is no abstract of this repor

A log-barrier method with Benders decomposition for solving two-stage stochastic linear programs

Mathematical Programming, Series B903507-536MPSB

Decomposition-based interior point methods for two-stage stochastic convex quadratic programs with recourse

Zhao [28] recently showed that the log barrier associated with the recourse function of two-stage stochastic linear programs behaves as a strongly self-concordant barrier and forms a self concordant family on the first stage solutions. In this paper we show that the recourse function is also strongly self-concordant and forms a self concordant family for the two-stage stochastic convex quadratic programs with recourse. This allows us to develop Benders decomposition based linearly convergent interior point algorithms. An analysis of such an algorithm is given in this paper.[28] G. Zhao: A log-barrier method with Benders decomposition for solving two-stage stochastic linear programs, Mathematical Programming Ser. A 90, (2001) 507-536

Two-Stage Stochastic Mixed Integer Linear Optimization

The primary focus of this dissertation is on optimization problems that involve uncertainty unfolding over time. In many real-world decisions, the decision-maker has to decide in the face of uncertainty. After the outcome of the uncertainty is observed, she can correct her initial decision by taking some corrective actions at a later time stage. These problems are known as stochastic optimization problems with recourse. In the case that the number of time stages is limited to two, these problems are referred to as two-stage stochastic optimization problems. We focus on this class of optimization problems in this dissertation. The optimization problem that is solved before the realization of uncertainty is called the first-stage problem and the problem solved to make a corrective action on the initial decision is called the second-stage problem. The decisions made in the second- stage are affected by both the first-stage decisions and the realization of random variables. Consequently, the two-stage problem can be viewed as a parametric optimization problem which involves the so-called value function of the second-stage problem. The value function describes the change in optimal objective value as the right-hand side is varied and understanding it is crucial to developing solution methods for two-stage optimization problems.In the first part of this dissertation, we study the value function of a MILP. We review the structural properties of the value function and its construction methods. We con- tribute by proposing a discrete representation of the MILP value function. We show that the structure of the MILP value function arises from two other optimization problems that are constructed from its discrete and continuous components. We show that our representation can explain certain structural properties of the MILP value function such as the sets over which the value function is convex. We then provide a simplification of the Jeroslow Formula obtained by applying our results. Finally, we describe a cutting plane algorithm for its construction and determine the conditions under which the pro- posed algorithm is finite.Traditionally, the solution methods developed for two-stage optimization problems consider the problem where the second-stage problem involves only continuous variables. In the recent years, however, two-stage problems with integer variables in the second- stage have been visited in several studies. These problems are important in practice and arise in several applications in supply chain, finance, forestry and disaster management, among others. The second part of this dissertation concerns the development and implementation of a solution method for the two-stage optimization problem where both the first and second stage involve mixed integer variables. We describe a generalization of the classical Benders’ method for solving mixed integer two-stage stochastic linear optimization problems. We employ the strong dual functions encoded in the branch-and-bound trees resulting from solution of the second-stage problem. We show that these can be used effectively within a Benders’ framework and describe a method for obtaining all required dual functions from a single, continuously refined branch-and-bound tree that is used to warm start the solution procedure for each subproblem.Finally, we provide details on the implementation of our proposed algorithm. The implementation allows for construction of several approximations of the value function of the second-stage problem. We use different warm-starting strategies within our proposed algorithm to solve the second-stage problems, including solving all second-stage problems with a single tree. We provide computational results on applying these strategies to the stochastic server problems (SSLP) from the stochastic integer programming test problem library (SIPLIB)

Spatial Energy System Modelling under Uncertainty with application to Thailand

The current awareness of the depletion in the fossil fuels reserves and the effect of green house gases (GHG) toward global warming has motivated many researchers in the area of energy system modelling. This thesis presents mathematical models to aid decision makers in determining the optimal spatially aggregated energy supply chain network to satisfy the future energy demand at the national level. Firstly, the energy planning problem using Thailand’s energy system as the case study is addressed by the development of a multi-period environmentally conscious deterministic energy system optimisation model. The model is formulated as a linear programming (LP) model that can address decision-making of the optimal future energy supply chain network at the national level with consideration of the scale of GHG emissions of the network. The determination of data required for the development of the proposed model is also tackled. Secondly, the reformulation of the multi-period deterministic model as a three-staged stochastic energy system optimisation model that can support decision-making under uncertainty in energy demand is addressed. Further extensions to the deterministic model include its reformulation to take into account the geographical location of an energy system. The linear programming model is reformulated as a mixed integer linear programming model (MILP) that can incorporated the spatial nature of the energy system as part of the decision-making process. The decisions to be determined include: (1) scale, type and location of energy production facility, (2) scale and type of resource usage in each location, (3) flow of resources and energy between grids to satisfy the energy demand throughout the planning horizon. Next, the Biomass-to-Energy supply chain network over long-term planning with application to Thailand is focused, based on the spatial MILP formulation. A higher complexity of geographical location is addressed as well as increases in types of biomass and biomass thermal conversion technologies. The objective function is modified to maximise the total network profit rather than minimising the total network costs. Finally, the long-term planning of a Waste-to Energy supply chain network with application to Thailand is investigated. The Waste-to-Energy system is addressed in view of investors as decision-makers as the objective function is also to maximise the total profit of the network. Different network structures of converting waste into energy are applied. The problem is also formulated as a MILP problem. This thesis reveals that, based on the model assumptions, the optimal environmentally conscious energy supply chain networks rely heavily on the utilisation of renewable resources throughout the country. With the abundant amount of biomass and waste resources available in Thailand, Biomass and Waste-to-Energy projects have a high potential in diversifying the use of fossil fuels as primary energy sources in Thailand

Two-Stage Stochastic Semidefinite Programming: Theory, Algorithms, and Application to AC Power Flow under Uncertainty

In real life decision problems, one almost always is confronted with uncertainty and risk. For practical optimization problems this is manifested by unknown parameters within the input data, or, an inexact knowledge about the system description itself. In case the uncertain problem data is governed by a known probability distribution, stochastic programming offers a variety of models hedging against uncertainty and risk. Most widely employed are two-stage models, who admit a recourse structure: The first-stage decisions are taken before the random event occurs. After its outcome, a recourse (second-stage) action is made, often but not always understood as some "compensation''. In the present thesis, the optimization problems that involve parameters which are not known with certainty are semidefinite programming problems. The constraint sets of these optimization problems are given by intersections of the cone of symmetric, positive semidefinite matrices with either affine or more general equations. Objective functions, formally, may be fairly general, although they often are linear as in the present thesis. We consider risk neutral and risk averse two-stage stochastic semidefinite programs with continuous and mixed-integer recourse, respectively. For these stochastic optimization problems we analyze their structure, derive solution methods relying on decomposition, and finally apply our results to unit commitment in alternating current (AC) power systems. Furthermore, deterministic unit commitment in AC power transmission systems is addressed. Beside traditional unit commitment constraints, the physics of power flow are included. To gain globally optimal solutions a recent semidefinite programming (SDP) approach is used which leads to large-scale semidefinite programs with discrete variables on top. As even the SDP relaxation of these programs is too large for being handled in an all-at-once manner by general SDP solvers, it requires an efficient and reliable method to tackle them. To this end, an algorithm based on Benders decomposition is proposed. With power demand (load) and in-feed from renewables serving as sources of uncertainty, two-stage stochastic programs are set up heading for unit commitment schedules which are both cost-effective and robust with respect to data perturbations. The impact of different, risk neutral and risk averse, stochastic criteria on the shapes of the optimal stochastic solutions will be examined. To tackle the resulting two-stage programs, we propose to approximate AC power flow by semidefinite relaxations. This leads to two-stage stochastic mixed-integer semidefinite programs having a special structure. To solve the latter, the L-shaped method and dual decomposition have been applied and compared.Betrachtet man reale Entscheidungsprobleme, die also der Wirklichkeit entstammen, so ist man fast immer mit Unsicherheiten und Risiken konfrontiert. Für konkrete Optimierungsprobleme äußert sich dies sowohl in Form von ungewissen Parametern in den Eingangsdaten, als auch durch eine unzureichende Kenntnis über die Systembeschreibung selbst. Handelt es sich um zufallsbehaftete Eingangsdaten, dessen Verteilung bekannt ist, so stellt die Stochastische Optimierung eine Vielzahl von Modellen bereit - allesamt mit dem Ziel sich gegen Unsicherheiten und Risiken abzusichern. Die am Häufigsten verwendeten stochastischen Modelle sind zweistufige Modelle. Diese gestatten folgende Kompensationsstrategie: Eine Erststufenentscheidung wird getroffen bevor das Zufallsereignis eintritt. Nach Realisierung des Zufalls können Korrekturmaßnahmen (zweite Stufe) ergriffen werden, welche häufig, aber nicht immer, als "Kompensation" verstanden werden. Die vorliegende Arbeit behandelt Semidefinite Programme, dessen Parameter nicht mit Sicherheit bekannt sind. Der Zulässigkeitsbereich dieser Optimierungsprobleme entsteht aus dem Durchschnitt affiner oder auch allgemeinerer Gleichungen mit dem Kegel der symmetrisch und positiv semidefiniten Matrizen. Die Zielfunktion kann relativ allgemein sein, wird aber häufig, wie es auch in dieser Arbeit der Fall ist, als linear angenommen. Es werden risikoneutrale und risikoaverse zweistufige stochastische semidefinite Optimierungsprobleme mit jeweils stetiger und gemischt-ganzzahliger Kompensation betrachtet. Wir analysieren die Struktur dieser stochastischen Optimierungsprobleme, leiten dekompositionsbasierte Lösungsverfahren her und wenden unsere Resultate auf das Problem der optimalen Kraftwerkseinsatzplanung in Wechselstromnetzen an. Ferner beschäftigt sich diese Arbeit mit der deterministischen Kraftwerkseinsatzplanung in Wechselstromnetzen. Neben den traditionellen technischen Bedingungen an die einzelnen Kraftwerke wird auch die Physik des Wechselstroms berücksichtigt. Um global optimale Lösungen zu erhalten wird eine auf Semidefinite Programmierung (SDP) basierende Lösungsstrategie benutzt. Dieser Ansatz resultiert in einem umfangreichen semidefiniten Programm, welches zusätzlich diskrete Entscheidungsvariablen enthält. Da selbst die SDP Relaxierung dieses Optimierungsproblems zu groß ist um es mittels gängiger SDP Löser auf einmal zu lösen, wird eine effiziente und zuverlässige Methode benötigt. Es wird ein Algorithmus basierend auf dem Dekompositionsprinzip von Benders vorgeschlagen. Ausgehend vom Energiebedarf (Last) und der Einspeisung der erneuerbaren Energien als Unsicherheitsquelle, wird ein zweistufiges stochastisches Optimierungsproblem formuliert. Das Ziel ist es, einen Kraftwerkseinsatzplan zu finden, der wirtschaftlich effektiv und robust gegenüber Veränderungen in den Daten ist. Es werden die Auswirkungen des risikoneutralen und risikoaversen Ansatzes auf die stochastische Lösung untersucht und miteinander verglichen. Um die resultierenden zweistufigen Programme zu lösen wird das Wechselstromnetz mit Hilfe des SDP Ansatzes approximiert. Dies führt zu zweistufigen stochastischen gemischt-ganzzahligen semidefiniten Programmen mit spezieller Struktur. Als Lösungsmethoden wurden die L-shaped Methode und die duale Dekomposition verwendet