61 research outputs found

    A Primal-Dual Augmented Lagrangian Penalty-Interior-Point Algorithm for Nonlinear Programming

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    This thesis treats a new numerical solution method for large-scale nonlinear optimization problems. Nonlinear programs occur in a wide range of engineering and academic applications like discretized optimal control processes and parameter identification of physical systems. The most efficient and robust solution approaches for this problem class have been shown to be sequential quadratic programming and primal-dual interior-point methods. The proposed algorithm combines a variant of the latter with a special penalty function to increase its robustness due to an automatic regularization of the nonlinear constraints caused by the penalty term. In detail, a modified barrier function and a primal-dual augmented Lagrangian approach with an exact l2-penalty is used. Both share the property that for certain Lagrangian multiplier estimates the barrier and penalty parameter do not have to converge to zero or diverge, respectively. This improves the conditioning of the internal linear equation systems near the optimal solution, handles rank-deficiency of the constraint derivatives for all non-feasible iterates and helps with identifying infeasible problem formulations. Although the resulting merit function is non-smooth, a certain step direction is a guaranteed descent. The algorithm includes an adaptive update strategy for the barrier and penalty parameters as well as the Lagrangian multiplier estimates based on a sensitivity analysis. Global convergence is proven to yield a first-order optimal solution, a certificate of infeasibility or a Fritz-John point and is maintained by combining the merit function with a filter or piecewise linear penalty function. Unlike the majority of filter methods, no separate feasibility restoration phase is required. For a fixed barrier parameter the method has a quadratic order of convergence. Furthermore, a sensitivity based iterative refinement strategy is developed to approximate the optimal solution of a parameter dependent nonlinear program under parameter changes. It exploits special sensitivity derivative approximations and converges locally with a linear convergence order to a feasible point that further satisfies the perturbed complementarity condition of the modified barrier method. Thereby, active-set changes from active to inactive can be handled. Due to a certain update of the Lagrangian multiplier estimate, the refinement is suitable in the context of warmstarting the penalty-interior-point approach. A special focus of the thesis is the development of an algorithm with excellent performance in practice. Details on an implementation of the proposed primal-dual penalty-interior-point algorithm in the nonlinear programming solver WORHP and a numerical study based on the CUTEst test collection is provided. The efficiency and robustness of the algorithm is further compared to state-of-the-art nonlinear programming solvers, in particular the interior-point solvers IPOPT and KNITRO as well as the sequential quadratic programming solvers SNOPT and WORHP

    Using the primal-dual interior point algorithm within the branch-price-and-cut method

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    AbstractBranch-price-and-cut has proven to be a powerful method for solving integer programming problems. It combines decomposition techniques with the generation of both columns and valid inequalities and relies on strong bounds to guide the search in the branch-and-bound tree. In this paper, we present how to improve the performance of a branch-price-and-cut method by using the primal-dual interior point algorithm. We discuss in detail how to deal with the challenges of using the interior point algorithm with the core components of the branch-price-and-cut method. The effort to overcome the difficulties pays off in a number of advantageous features offered by the new approach. We present the computational results of solving well-known instances of the vehicle routing problem with time windows, a challenging integer programming problem. The results indicate that the proposed approach delivers the best overall performance when compared with a similar branch-price-and-cut method which is based on the simplex algorithm

    Active-set prediction for interior point methods\ud using controlled perturbations

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    We propose the use of controlled perturbations to address the challenging question of optimal active-set prediction for interior point methods. Namely, in the context of linear programming, we consider perturbing the inequality constraints/bounds so as to enlarge the feasible set. We show that if the perturbations are chosen appropriately, the solution of the original problem lies on or close to the central path of the perturbed problem. We also nd that a primal-dual path-following algorithm applied to the perturbed problem is able to accurately predict the optimal active set of the original problem when the duality gap for the perturbed problem is not too small; furthermore, depending on problem conditioning, this prediction can happen sooner than predicting the active-set for the perturbed problem or for the original one if no perturbations are used. Encouraging preliminary numerical experience is reported when comparing activity prediction for the perturbed and unperturbed problem formulations

    Advanced Optimization and Statistical Methods in Portfolio Optimization and Supply Chain Management

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    This dissertation is on advanced mathematical programming with applications in portfolio optimization and supply chain management. Specifically, this research started with modeling and solving large and complex optimization problems with cone constraints and discrete variables, and then expanded to include problems with multiple decision perspectives and nonlinear behavior. The original work and its extensions are motivated by real world business problems.The first contribution of this dissertation, is to algorithmic work for mixed-integer second-order cone programming problems (MISOCPs), which is of new interest to the research community. This dissertation is among the first ones in the field and seeks to develop a robust and effective approach to solving these problems. There is a variety of important application areas of this class of problems ranging from network reliability to data mining, and from finance to operations management.This dissertation also contributes to three applications that require the solution of complex optimization problems. The first two applications arise in portfolio optimization, and the third application is from supply chain management. In our first study, we consider both single- and multi-period portfolio optimization problems based on the Markowitz (1952) mean/variance framework. We have also included transaction costs, conditional value-at-risk (CVaR) constraints, and diversification constraints to approach more realistic scenarios that an investor should take into account when he is constructing his portfolio. Our second work proposes the empirical validation of posing the portfolio selection problem as a Bayesian decision problem dependent on mean, variance and skewness of future returns by comparing it with traditional mean/variance efficient portfolios. The last work seeks supply chain coordination under multi-product batch production and truck shipment scheduling under different shipping policies. These works present a thorough study of the following research foci: modeling and solution of large and complex optimization problems, and their applications in supply chain management and portfolio optimization.Ph.D., Business Administration -- Drexel University, 201

    Column-generation and interior point methods applied to the long-term electric power-planning problem

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    Aquesta tesi s'adreça al problema de planificació de la generació elèctrica a llarg termini per a una companyia específica (SGC) que participa en un mercat liberalitzat organitzat en un pool. Els objectius de la tesi són: modelitzar aquest problema, i desenvolupar i implementar tècniques apropiades i eficients que el resolguin. Un planificació òptima a llarg termini és important, per exemple, per a la confecció de pressupostos, o per a la gestió de compres/consum de combustibles. Una altra aplicació és la de guiar la planificació a curt termini perquè aquesta tingui en compte decisions preses sota una òptica de llarg termini. La nostra proposta per a fer la planificació de la generació és optimitzar la generació esperada de cada unitat (o la unió de diverses unitats de característiques semblants) del pool per a cada interval en que dividim el llarg termini. El model bàsic per la planificació de la generació a llarg termini (LTGP) maximitza el benefici de totes les unitats del pool. La constricció més important és la satisfacció de la demanda, ja que el sistema està sempre balancejat. Utilitzem la formulació de Bloom i Gallant, la qual modela la càrrega a través d'una monòtona de càrrega per cada interval i requereix un número exponencial de constriccions lineals de desigualtat, anomenades LMCs. Altres constriccions (lineals) incloses en el model són: garantia de potència, límits en la disponibilitat de combustibles, emissions màximes de CO2 o una quota de mercat mínima per a la SGC. Una extensió d'aquest model és la planificació conjunta de l'assignació de manteniments de les unitats tèrmiques d'una SGC amb la planificació de la generació. El model conjunt és un problema quadràtic amb variables binàries i contínues. Per resoldre aquest model es proposa un parell d'heurístiques i s'ha implementat un prototipus de branch and bound en AMPL.Aquesta tesi també proposa una manera per coordinar el model LTGP proposat amb una planificació a curt termini. Es desenvolupa un model de curt que inclou els resultats de llarg termini. Donat que el model de planificació a llarg termini s'ha de resoldre sovint (principalment per passar informació acurada al model de curt), les tècniques emprades per a resoldre'l han de donar resultats fiables en un espai de temps curt. Les tècniques aplicades han estat:· Donat que les constriccions de recobriment i les fites de no negativitat defineixen un políedre convex els vèrtexs del qual són fàcils de trobar el model es transforma i les variables esdevenen els coeficients convexos que defineixen un punt. Aquest nou problema es resolt amb l'algoritme de Murtagh i Saunders, que és un procediment òptim. Aquest algoritme s'aplica sota un esquema de generació de columnes donat que el número de vèrtexs del políedre és comparable al número de constriccions. L'avantatge d'aquest mètode és que els vèrtexs es van generant a mesura que es necessiten.· L'aplicació de mètodes directes és computacionalment costós donat el número exponencial de LMCs. De totes maneres, a l'òptim només un conjunt reduït de constriccions de recobriment seran actives. Hem desenvolupat una heurística, anomenada heurística GP, la qual genera un subconjunt de constriccions, entre les quals hi ha les LMCs que són actives a l'òptim. L'heurística resol una seqüència de problemes quadràtics, els quals incrementen el número de LMCs considerades a cada iteració. Els problemes es resolen amb mètodes de punt interior que s'inicialitzen amb tècniques de warm start per tal d'accelerar la convergència cap a la nova solució. Aquest procediment resulta ser molt més eficient que el de generació de columnes. La modelització i els casos de prova estan basats en dades d'un sistema de pool pur i de mercat com ha estat a Espanya fins el juliol de 2006.This thesis presents an approach to the long-term planning of power generation for a company (SGC) participating in a liberalized market organized as a pool. The goal of this thesis is two-fold: to model the problem and to develop and implement appropriate and efficient techniques for solving it.The optimization of the long-term generation planning is important for budgeting and planning fuel acquisitions, and to give a frame where to fit short-term generation planning.Our proposal for planning long-term generation is to optimize the expected generation of each unit (or the merger of several units of the same type) in the power pool over each interval into which the long-term horizon is split.The basic model for long-term generation planning (LTGP) maximizes the profit for all the units participating in the pool. The most important constraint is matching demand, since the market always clears. The Bloom and Gallant formulation is used, which models the load with a load-duration curve for each interval and requires an exponential number of linear inequality constraints, called herein LMCs. Other (linear) constraints included in the model are: minimum generation time, limits on the availability of fuel, maximum CO2 emission limits or the market share of the SGC. This thesis also proposes the way in which coordination between the LTGP model developed and a short-term plan should be considered and provides a model for short-term electrical power planning adapted to the LTGP proposed and which includes the long-term results.Another decision that needs to be taken from a long-term point of view is the joint scheduling of thermal unit maintenances with the generation planning of a particular SGC. The results of a prototype of a Branch and Bound implemented in AMPL are included in this thesis.Long-term planning needs to be considered before short-term planning and whenever the real situation deviates from the forecasted parameters, so the techniques implemented must be efficient so as to provide reliable solutions in a short time. Two methods for handling the LMCs are proposed and compared:● A decomposition technique exploits the fact that the LMCs plus the non-negativity bounds define a convex polyhedron for each interval whose vertices are easy to find. Thus, the problem is transformed and the variables become the coefficients of a convex combination of the vertices. The transformed problem is quadratic with linear constraints, making it suitable to be solved with the Murtagh & Saunders algorithm, which gives an optimal solution. A column-generation approach is used because the number of vertices of the polyhedron is comparable to the number of LMCs. The advantage of this method is that it does not require previous computation of all of the vertices, but rather computes them as the algorithm iterates.● The application of direct methods is computationally difficult because of the exponential number of inequality LMCs. However, only a reduced subset of LMCs will be active at the optimizer. A heuristic, named GP heuristic, has been devised which is able to find a reduced set of LMCs including those that are active at the optimizer. It solves a sequence of quadratic problems in which the set of LMCs considered is enlarged at each iteration. The quadratic problems are solved with an interior point method, and warm starts are employed to accelerate the solution of the successively enlarged quadratic problems. This procedure is more efficient than the column generation one.The modeling and tests of this thesis are based on the pure pool system and market data from the Spanish system up to July 2006

    A primal-dual interior-point relaxation method with adaptively updating barrier for nonlinear programs

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    Based on solving an equivalent parametric equality constrained mini-max problem of the classic logarithmic-barrier subproblem, we present a novel primal-dual interior-point relaxation method for nonlinear programs. In the proposed method, the barrier parameter is updated in every step as done in interior-point methods for linear programs, which is prominently different from the existing interior-point methods and the relaxation methods for nonlinear programs. Since our update for the barrier parameter is autonomous and adaptive, the method has potential of avoiding the possible difficulties caused by the unappropriate initial selection of the barrier parameter and speeding up the convergence to the solution. Moreover, it can circumvent the jamming difficulty of global convergence caused by the interior-point restriction for nonlinear programs and improve the ill conditioning of the existing primal-dual interiorpoint methods as the barrier parameter is small. Under suitable assumptions, our method is proved to be globally convergent and locally quadratically convergent. The preliminary numerical results on a well-posed problem for which many line-search interior-point methods fail to find the minimizer and a set of test problems from the CUTE collection show that our method is efficient.Comment: submitted to SIOPT on April 14, 202
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