<p>In this thesis, we extend the Ai-Zhang direction to the class of semidefinite optimization problems. We define a new wide neighborhood N(τ1 ,τ2 ,η) and, as usual, we utilize symmetric directions by scaling the Newton equation with special matrices. After defining the "positive part" and the "negative part" of a symmetric matrix, we solve the Newton equation with its right hand side replaced first by its positive part and then by its negative part, respectively. In this way, we obtain a decomposition of the usual Newton direction and use different step lengths for each of them.</p><p>Starting with a feasible point (X^0 , y^0 , S^0) in N(τ1, τ2 , η ), the algorithm terminates in at most 0(η√( κ∞n)log(1/ε) iterations, where κ∞ is a parameter associated with the scaling matrix and ε is the required precision. To our best knowledge, when the parameter η is a constant, this is the first large neighborhood path-following Interior Point Method (IPM) with the same complexity as small neighborhood path-following IPMs for semidefinite optimization that use the Nesterov-Todd direction. In the case when η is chosen to be in the order of √η, our complexity bound coincides with the known bound for classical large neighborhood IPMs.</p><p>To make this thesis more accessible to readers who are new in this area, we start with a brief introduction to IPMs and SDO. The basic concepts and principles of IPMs and SDO are presented in Chapter 2 and 3.</p> / Thesis / Master of Applied Science (MASc)

Over the last 25 years, interior-point methods (IPMs) have emerged as a viable class of algorithms for solving various forms of conic optimization problems. Most IPMs use a modified Newton method to determine the search direction at each iteration. The system of equations corresponding to the modified Newton system can often be reduced to the so-called normal equation, a system of equations whose matrix ADA' is positive definite, yet often ill-conditioned. In this thesis, we first investigate the theoretical properties of the maximum weight basis (MWB) preconditioner, and show that when applied to a matrix of the form ADA', where D is positive definite and diagonal, the MWB preconditioner yields a preconditioned matrix whose condition number is uniformly bounded by a constant depending only on A. Next, we incorporate the results regarding the MWB preconditioner into infeasible, long-step, primal-dual, path-following algorithms for linear programming (LP) and convex quadratic programming (CQP). In both LP and CQP, we show that the number of iterative solver iterations of the algorithms can be uniformly bounded by n and a condition number of A, while the algorithmic iterations of the IPMs can be polynomially bounded by n and the logarithm of the desired accuracy. We also expand the scope of the LP and CQP algorithms to incorporate a family of preconditioners, of which MWB is a member, to determine an approximate solution to the normal equation. For the remainder of the thesis, we develop a new preconditioning strategy for solving systems of equations whose associated matrix is positive definite but ill-conditioned. Our so-called adaptive preconditioning strategy allows one to change the preconditioner during the course of the conjugate gradient (CG) algorithm by post-multiplying the current preconditioner by a simple matrix, consisting of the identity matrix plus a rank-one update. Our resulting algorithm, the Adaptive Preconditioned CG (APCG) algorithm, is shown to have polynomial convergence properties. Numerical tests are conducted to compare a variant of the APCG algorithm with the CG algorithm on various matrices.

This research studies how to efficiently predict optimal active constraints of an inequality constrained optimization problem, in the context of Interior Point Methods (IPMs). We propose a framework based on shifting/perturbing the inequality constraints of the problem. Despite being a class of powerful tools for solving Linear Programming (LP) problems, IPMs are well-known to encounter difficulties with active-set prediction due essentially to their construction. When applied to an inequality constrained optimization problem, IPMs generate iterates that belong to the interior of the set determined by the constraints, thus avoiding/ignoring the combinatorial aspect of the solution. This comes at the cost of difficulty in predicting the optimal active constraints that would enable termination, as well as increasing ill-conditioning of the solution process. We show that, existing techniques for active-set prediction, however, suffer from difficulties in making an accurate prediction at the early stage of the iterative process of IPMs; when these techniques are ready to yield an accurate prediction towards the end of a run, as the iterates approach the solution set, the IPMs have to solve increasingly ill-conditioned and hence difficult, subproblems. To address this challenging question, we propose the use of controlled perturbations. Namely, in the context of LP problems, we consider perturbing the inequality constraints (by a small amount) so as to enlarge the feasible set. We show that if the perturbations are chosen judiciously, the solution of the original problem lies on or close to the central path of the perturbed problem. We solve the resulting perturbed problem(s) using a path-following IPM while predicting on the way the active set of the original LP problem; we find that our approach is able to accurately predict the optimal active set of the original problem before the duality gap for the perturbed problem gets 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. Proof-of-concept algorithms are presented and encouraging preliminary numerical experience is also reported when comparing activity prediction for the perturbed and unperturbed problem formulations. We also extend the idea of using controlled perturbations to enhance the capabilities of optimal active-set prediction for IPMs for convex Quadratic Programming (QP) problems. QP problems share many properties of LP, and based on these properties, some results require more care; furthermore, encouraging preliminary numerical experience is also presented for the QP case.

The aim of this research is to demonstrate some more efficient approaches to solve the n-1 security constrained optimal power flow (SCOPF) problems by using structure-exploiting primal-dual interior point methods (IPM). Firstly, we consider a DC-SCOPF model, which is a linearized version of AC-SCOPF. One new reformulation of the DC-SCOPF model is suggested, in which most matrices that need to be factorized are constant. Consequently, most numerical factorizations and a large number of back-solve operations only need to be performed once throughout the entire IPM process. In the framework of the structure-exploiting IPM implementation, one of the major computational efforts consists of forming the Schur complement matrix, which is very computationally expensive if no further measure is applied. One remedy is to apply a preconditioned iterative method to solve the corresponding linear systems which appear in assembling the Schur complement matrix. We suggest two main schemes to pick a good and robust preconditioner for SCOPF problems based on combining different “active” contingency scenarios. The numerical results show that our new approaches are much faster than the default structure-exploiting method in OOPS, and also that it requires less memory. The second part of this thesis goes to the standard AC-SCOPF problem, which is a nonlinear and nonconvex optimization problem. We present a new contingency generation algorithm: it starts with solving the basic OPF problem, which is a much smaller problem of the same structure, and then generates contingency scenarios dynamically when needed. Some theoretical analysis of this algorithm is shown for the linear case, while the numerical results are exciting, as this new algorithm works for both AC and DC cases. It can find all the active scenarios and significantly reduce the number of scenarios one needs to contain in the model. As a result, it speeds up the solving process and may require less IPM iterations. Also, some heuristic algorithms are designed and presented to predict the active contingencies for the standard AC-SCOPF, based on the use of AC-OPF or DC-SCOPF. We test our heuristic algorithms on the modified IEEE 24-bus system, and also present their corresponding numerical results in the thesis.

O objetivo deste trabalho consiste no desenvolvimento, estudo e implementação de métodos de pontos interiores específicos para o problema de planejamento do tratamento de câncer por radioterapia. Este é um problema de grande porte que contém uma estrutura matricial particular. A exploração desta estrutura de forma eficiente obtém bom desempenho computacional, através da redução da dimensão dos sistemas lineares que devem ser resolvidos a cada iteração, agilizando a definição de um tratamento adequado, uma vez que tipicamente várias simulações são realizadas antes da definição de um plano definitivo. Resultados numéricos em Matlab ilustram a eficiência desta abordagem em problemas reais e mostram a superioridade do método preditor corretor em comparação ao método primal-dual. / In this work, a specialized interior point method is developed for planning cancer treatment by radiotherapy. This is a large-scale problem with a specific matrix structure. That structure is explored in an efficient way reducing the dimension of the linear system which must be solved at each iteration speeding up the treatment design since usually several versions must be solved to obtain a satisfactory plan. Moreover, the system obtained is sparse, symmetric and positive definite. Numerical results in Matlab illustrate the efficiency of this approach in real problems and show the superiority of the preditor-corrector method in comparison to the primal-dual method.

<p>In order to design robust controllers, H-infinity synthesis is a common tool to use. The controllers that result from these algorithms are typically of very high order, which complicates implementation. However, if a constraint on the maximum order of the controller is set, that is lower than the order of the plant, the problem is no longer convex and it is then relatively hard to solve. These problems become very complex, even when the order of the system to be controlled is low.</p><p>The approach used in the thesis is based on formulating the constraint on the maximum order of the plant as a polynomial equation. By using the fact that the polynomial is non-negative on the feasible set, the problem is reformulated as an optimization problem where the nonconvex polynomial function is to be minimized over a convex set defined by linear matrix inequalities.</p><p>To solve this optimization problem, two methods have been proposed. The first method is a barrier method and the second one is a method based on a primal-dual framework. These methods have been evaluated on several problems and compared with a well-known method found in the literature. To motivate this choice of method, we have made a brief survey of available methods available for solving the same or related problems.</p><p>The proposed methods emerged as the best methods among the three for finding lower order controllers with the same or similar performance as the full order controller. When the aim is to find the lowest order controller with no worse than +50% increase in the closed loop H-infinity norm, then the three compared methods perform equally well.</p>

Mathematical program with equilibrium constraints (MPEC)has extensive applications in practical areas such as traffic control, engineering design, and economic modeling. Some generalized stationary points of MPEC are studied to better describe the limiting points produced by interior point methods for MPEC.A primal-dual interior point method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced without assuming strict complementarity or linear independence constraint qualification. Under very general assumptions, the algorithm can always find some point with strong or weak stationarity. In particular, it is shown that every limiting point of the generated sequence is a piece-wise stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a certain point with weak stationarity can be obtained. Preliminary numerical results are satisfactory, which include a case analyzed by Leyffer for which the penalty interior point algorithm failed to find a stationary solution. / Singapore-MIT Alliance (SMA)

We propose and test a new adaptive conjoint analysis method that draws on recent polyhedral “interior-point” developments in mathematical programming. The method is designed to offer accurate estimates after relatively few questions in problems involving many parameters. Each respondent’s ques-tions are adapted based upon prior answers by that respondent. The method requires computer support but can operate in both Internet and off-line environments with no noticeable delay between questions. We use Monte Carlo simulations to compare the performance of the method against a broad array of relevant benchmarks. While no method dominates in all situations, polyhedral algorithms appear to hold significant potential when (a) metric profile comparisons are more accurate than the self-explicated importance measures used in benchmark methods, (b) when respondent wear out is a concern, and (c) when product development and/or marketing teams wish to screen many features quickly. We also test hybrid methods that combine polyhedral algorithms with existing conjoint analysis methods. We close with suggestions on how polyhedral methods can be used to address other marketing problems. / Sloan School of Management and the Center for Innovation in Product Development at MIT

Many optimization problems can be formulated as second order cone programming (SOCP) problems. Theoretical results show that applying interior-point method (IPM) to SOCP has global polynomial convergence. However, various stability issues arise in the implementation of IPM. The standard normal equation based implementation of IPM encounters stability problems in the computation of search direction. In this paper, an augmented system approach is proposed to overcome the stability problems. Numerical experiments show that the new approach can improve the stability. / Singapore-MIT Alliance (SMA)

We present a practical algorithm for computing the minimum volume n-dimensional ellipsoid that must contain m given points al,...,am C Rn . This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interior-point and active-set method for solving this problem. Our computational results demonstrate that our method solves very large problem instances (m = 30, 000 and n = 30) to a high degree of accuracy in under 30 seconds on a personal computer.