Experimental investigations in combining primal dual interior point method and simplex based LP solvers

The use of a primal dual interior point method (PD) based optimizer as a robust linear programming (LP) solver is now well established. Instead of replacing the sparse simplex algorithm (SSX), the PD is increasingly seen as complementing it. The progress of PD iterations is not hindered by the degeneracy or the stalling problem of the SSX, indeed it reaches the 'near optimum' solution very quickly. The SSX algorithm, in contrast, is not affected by the boundary conditions which slow down the convergence of the PD. If the solution to the LP problem is non unique, the PD algorithm converges to an interior point of the solution set while the SSX algorithm finds an extreme point solution. To take advantage of the attractive properties of both the PD and the SSX, we have designed a hybrid framework whereby cross over from PD to SSX can take place at any stage of the PD optimization run. The cross over to SSX involves the partition of the PD solution set to active and dormant variables. In this paper we examine the practical difficulties in partitioning the solution set, we discuss the reliability of predicting the solution set partition before optimality is reached and report the results of combining exact and inexact prediction with SSX basis recovery

Modelling and solution methods for portfolio optimisation

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University, 16/01/2004.In this thesis modelling and solution methods for portfolio optimisation are presented. The investigations reported in this thesis extend the Markowitz mean-variance model to the domain of quadratic mixed integer programming (QMIP) models which are 'NP-hard' discrete optimisation problems. In addition to the modelling extensions a number of challenging aspects of solution algorithms are considered. The relative performances of sparse simplex (SSX) as well as the interior point method (IPM) are studied in detail. In particular, the roles of 'warmstart' and dual simplex are highlighted as applied to the construction of the efficient frontier which requires processing a family of problems; that is, the portfolio planning model stated in a parametric form. The method of solving QMIP models using the branch and bound algorithm is first developed; this is followed up by heuristics which improve the performance of the (discrete) solution algorithm. Some properties of the efficient frontier with discrete constraints are considered and a method of computing the discrete efficient frontier (DEF) efficiently is proposed. The computational investigation considers the efficiency and effectiveness in respect of the scale up properties of the proposed algorithm. The extensions of the real world models and the proposed solution algorithms make contribution as new knowledge

Topics in exact precision mathematical programming

The focus of this dissertation is the advancement of theory and computation related to exact precision mathematical programming. Optimization software based on floating-point arithmetic can return suboptimal or incorrect resulting because of round-off errors or the use of numerical tolerances. Exact or correct results are necessary for some applications. Implementing software entirely in rational arithmetic can be prohibitively slow. A viable alternative is the use of hybrid methods that use fast numerical computation to obtain approximate results that are then verified or corrected with safe or exact computation. We study fast methods for sparse exact rational linear algebra, which arises as a bottleneck when solving linear programming problems exactly. Output sensitive methods for exact linear algebra are studied. Finally, a new method for computing valid linear programming bounds is introduced and proven effective as a subroutine for solving mixed-integer linear programming problems exactly. Extensive computational results are presented for each topic.Ph.D.Committee Chair: Dr. William J. Cook; Committee Member: Dr. George Nemhauser; Committee Member: Dr. Robin Thomas; Committee Member: Dr. Santanu Dey; Committee Member: Dr. Shabbir Ahmed; Committee Member: Dr. Zonghao G

Modelling and solution methods for portfolio optimisation

In this thesis modelling and solution methods for portfolio optimisation are presented. The investigations reported in this thesis extend the Markowitz mean-variance model to the domain of quadratic mixed integer programming (QMIP) models which are 'NP-hard' discrete optimisation problems. In addition to the modelling extensions a number of challenging aspects of solution algorithms are considered. The relative performances of sparse simplex (SSX) as well as the interior point method (IPM) are studied in detail. In particular, the roles of 'warmstart' and dual simplex are highlighted as applied to the construction of the efficient frontier which requires processing a family of problems; that is, the portfolio planning model stated in a parametric form. The method of solving QMIP models using the branch and bound algorithm is first developed; this is followed up by heuristics which improve the performance of the (discrete) solution algorithm. Some properties of the efficient frontier with discrete constraints are considered and a method of computing the discrete efficient frontier (DEF) efficiently is proposed. The computational investigation considers the efficiency and effectiveness in respect of the scale up properties of the proposed algorithm. The extensions of the real world models and the proposed solution algorithms make contribution as new knowledge.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Solving Saddle Point Formulations of Linear Programs with Frank-Wolfe

The problem of solving a linear program (LP) is ubiquitous in industry, yet in recent years the size of linear programming problems has grown and continues to do so. State-of-the-art LP solvers make use of the Simplex method and primal-dual interior-point methods which are able to provide accurate solutions in a reasonable amount of time for most problems. However, both the Simplex method and interior-point methods require solving a system of linear equations at each iteration, an operation that does not scale well with the size of the problem. In response to the growing size of linear programs and poor scalability of existing algorithms, researchers have started to consider first-order methods for solving large scale linear programs. The best known first-order method for general linear programming problems is PDLP. First-order methods for linear programming are characterized by having a matrix-vector product as their primary computational cost. We present a first-order primal-dual algorithm for solving saddle point formulations of linear programs, named FWLP (Frank-Wolfe Linear Programming). We provide some theoretical results regarding the behavior of our algorithm, however no convergence guarantees are provided. Numerical investigations suggest that our algorithm has error O(1/sqrt(k)) after k iterations, worse than that of PDLP, however we show that our algorithm has advantages for solving very large LPs in practice such as only needing part of the matrix A at each iteration

Distributed multi-agent pathfinding in horizontal transportation

Horizontal transportation in maritime container terminals plays a crucial role in ensuring safe, efficient, and cost-effective operations. Heavy working machines, such as straddle carriers, trucks, and automated guided vehicles, transport containers between cranes, creating complex routing problems known as multi-agent pathfinding (MAPF) problems. Existing solutions may not adequately address the unique challenges presented by container terminals, necessitating the development of new algorithms. This thesis aims to develop and demonstrate a distributed MAPF algorithm for horizontal transportation in container terminals. The MAPF problem is first formulated as a binary linear programming (BLP) model by expressing the actions in the container terminal using a directed pseudograph. Optimal solutions are obtained using PYOMO, an open-source Python-based optimization software. The Augmented Lagrangian, a graph pathfinding algorithm, and a stochastic element are then employed to create a sub-optimal, distributed algorithm. The developed algorithm is evaluated against an optimal solution and a reference method that prioritizes calculating the path for one agent at a time while taking into account previously calculated paths. A simulator is set up to emulate horizontal transportation in a maritime container terminal, by modeling the terminal as a graph in MATLAB. In the simulator, MAPF algorithms are applied in combination with a high-level coordinator assigning destinations. The experimental part of this thesis investigates the trade-off between solution time (iterations) and solution quality by tuning algorithm parameters and evaluating the performance of the distributed algorithm in comparison to the priority-based method under two different map layouts, particularly addressing the presence of a bottleneck. The results demonstrate the need to adapt the algorithm's parameters and strategies according to specific environments and map layouts, to ensure good performance across various scenarios. The main contribution of this thesis lies in the development of a adaptable, distributed MAPF solution that can ultimately address diverse scenarios and environments. Merikonttiterminaaleissa konttien vaakatasoinen kuljettaminen on keskeinen tekijä turvallisen, tehokkaan ja kustannustehokkaan toiminnan varmistamisessa. Raskaat työkoneet, kuten konttilukit, kuorma-autot ja automaattisti ohjatut ajoneuvot (AGV:t), kuljettavat kontteja nosturien välillä, luoden monimutkaisia reititysongelmia, joita kutsutaan monitoimija-reitinhaku (MAPF) -ongelmiksi. Olemassa olevat ratkaisut eivät riittävästi käsittele konttiterminaaleille asetettuja erityisiä haasteita, mikä edellyttää uusien hajautettujen MAPF-algoritmien kehittämistä. Tämän diplomityön tavoitteena on kehittää ja esitellä hajautettu MAPF-algoritmi vaakasuuntaiseen kuljetukseen konttiterminaaleissa. MAPF-ongelma mallinnetaan ensin binääriseksi lineaariseksi ohjelmointimalliksi (BLP), jossa konttiterminaalissa liikkuminen mallinnetaan suunnattuna pseudograafina. Optimaalisia ratkaisuja tutkitaan käyttämällä PYOMO-ohjelmistoa, joka on avoimen lähdekoodin Python-pohjainen optimointiohjelmisto. Tämän jälkeen laajennettua Lagrangen kertoimien menetelmää, graafin polunetsintä algoritmia ja stokastistista elementtiä käytetään luomaan lähes optimaalinen, hajautettavissa oleva algoritmi. Kehitettyä algoritmia arvioidaan vertaamalla sitä optimaaliseen ratkaisuun ja viite-menetelmään, joka priorisoi polkujen laskemista yhden agentin kerrallaan ottaen huomioon aiemmin lasketut polut. Simulaattori rakennetaan jäljittelemään vaakasuuntaista kuljetusta merikonttiterminaaleissa, mallintamalla terminaali graafina MATLABissa. Simulaattorissa MAPF-algoritmeja sovelletaan yhdessä korkean tason koordinaattorin kanssa, joka määrittelee määränpäät. Diplomityön kokeellisessa osuudessa tarkastellaan ratkaisuajan (iteraatioiden) ja ratkaisun laadun välistä tasapainoa kokeilemalla erilaisia parametreja ja arvioimalla hajautetun algoritmin suorituskykyä verrattuna prioriteettipohjaiseen menetelmään kahden erilaisen karttapohjan päällä, keskittyen erityisesti pullonkaulan lisäämisen tuomiin vaikutuksiin. Tulokset osoittavat, että algoritmin parametrien ja strategioiden mukauttaminen erityisiin ympäristöihin ja karttapohjiin on tarpeen, jos halutaan varmistua tulosten olevan mahdollisimman lähellä optimaalisuutta erilaisissa skenaarioissa. Tämän diplomityön pääasiallinen kontribuutio on joustavan, hajautetun MAPF-ratkaisun kehittäminen, jolla voidaan tulevaisuudessa käsitellä erilaisia tilanteita ja ympäristöjä

Experiments in reduction techniques for linear and integer programming

This study consisted of evaluating the relative performance to a selection of the most promising size-reduction techniques. Experiments and comparisons were made among these techniques on a series of tested problems to determine their relative efficiency, efficiency versus time etc. Three main new methods were developed by modifying and extending the previous ones. These methods were also tested and their results are compared with the earlier methods

Semidefinite Programming. methods and algorithms for energy management

La présente thèse a pour objet d explorer les potentialités d une méthode prometteuse de l optimisation conique, la programmation semi-définie positive (SDP), pour les problèmes de management d énergie, à savoir relatifs à la satisfaction des équilibres offre-demande électrique et gazier.Nos travaux se déclinent selon deux axes. Tout d abord nous nous intéressons à l utilisation de la SDP pour produire des relaxations de problèmes combinatoires et quadratiques. Si une relaxation SDP dite standard peut être élaborée très simplement, il est généralement souhaitable de la renforcer par des coupes, pouvant être déterminées par l'étude de la structure du problème ou à l'aide de méthodes plus systématiques. Nous mettons en œuvre ces deux approches sur différentes modélisations du problème de planification des arrêts nucléaires, réputé pour sa difficulté combinatoire. Nous terminons sur ce sujet par une expérimentation de la hiérarchie de Lasserre, donnant lieu à une suite de SDP dont la valeur optimale tend vers la solution du problème initial.Le second axe de la thèse porte sur l'application de la SDP à la prise en compte de l'incertitude. Nous mettons en œuvre une approche originale dénommée optimisation distributionnellement robuste , pouvant être vue comme un compromis entre optimisation stochastique et optimisation robuste et menant à des approximations sous forme de SDP. Nous nous appliquons à estimer l'apport de cette approche sur un problème d'équilibre offre-demande avec incertitude. Puis, nous présentons une relaxation SDP pour les problèmes MISOCP. Cette relaxation se révèle être de très bonne qualité, tout en ne nécessitant qu un temps de calcul raisonnable. La SDP se confirme donc être une méthode d optimisation prometteuse qui offre de nombreuses opportunités d'innovation en management d énergie.The present thesis aims at exploring the potentialities of a powerful optimization technique, namely Semidefinite Programming, for addressing some difficult problems of energy management. We pursue two main objectives. The first one consists of using SDP to provide tight relaxations of combinatorial and quadratic problems. A first relaxation, called standard can be derived in a generic way but it is generally desirable to reinforce them, by means of tailor-made tools or in a systematic fashion. These two approaches are implemented on different models of the Nuclear Outages Scheduling Problem, a famous combinatorial problem. We conclude this topic by experimenting the Lasserre's hierarchy on this problem, leading to a sequence of semidefinite relaxations whose optimal values tends to the optimal value of the initial problem.The second objective deals with the use of SDP for the treatment of uncertainty. We investigate an original approach called distributionnally robust optimization , that can be seen as a compromise between stochastic and robust optimization and admits approximations under the form of a SDP. We compare the benefits of this method w.r.t classical approaches on a demand/supply equilibrium problem. Finally, we propose a scheme for deriving SDP relaxations of MISOCP and we report promising computational results indicating that the semidefinite relaxation improves significantly the continuous relaxation, while requiring a reasonable computational effort.SDP therefore proves to be a promising optimization method that offers great opportunities for innovation in energy management.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF