Thesis: M.S., Massachusetts Institute of Technology, Sloan School of Management, 1980 / Bibliography: leaf 53. / by Jai Yong Oum. / M.S. / M.S. Massachusetts Institute of Technology, Sloan School of Management

<p> In this thesis parametric analysis for conic quadratic optimization problems is studied. In parametric analysis, which is often referred to as parametric optimization or parametric programming, a perturbation parameter is introduced into the optimization problem, which means that the coefficients in the objective function of the problem and in the right-hand-side of the constraints are perturbed. First, we describe linear, convex quadratic and second order cone optimization problems and their parametric versions. Second, the theory for finding solutions of the parametric problems is developed. We also present algorithms for solving such problems. Third, we demonstrate how to use parametric optimization techniques to solve multiobjective optimization problems and compute Pareto efficient surfaces. </p> <p> We implement our novel algorithm for hi-parametric quadratic optimization. It utilizes existing solvers to solve auxiliary problems. We present numerical results produced by our parametric optimization package on a number of practical financial and non-financial computational problems. In the latter we consider problems of drug design and beam intensity optimization for radiation therapy. </p> <p> In the financial applications part, two risk management optimization models are developed or extended. These two models are a portfolio replication framework and a credit risk optimization framework. We describe applications of multiobjective optimization to existing financial models and novel models that we have developed. We solve a number of examples of financial multiobjective optimization problems using our parametric optimization algorithms. </p> / Thesis / Doctor of Philosophy (PhD)

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In this paper, for a select group of quadratic algebras, we investigate restrictions necessary on the generators of the ideal for the resulting algebra to be Koszul. Techniques include the use of Gröbner bases and development of Koszul resolutions. When the quadratic algebra is Koszul, we provide the associated linear resolution of the field. When not Koszul, we describe the maps of the resolution up to the instance of nonlinearity. / Ph. D.

The objective of this work is to solve a model one dimensional duct design problem using a particular optimization method. The design problem is formulated as an equality constrained optimization, called All at once method, so that the analysis problem is not solved until the optimal design is reached. Furthermore, the block structure in the Jacobian of the linearized constraints is exploited by decomposing the variables into the design and flow parts. To achieve this, Sequential quadratic programming with BFGS update for the reduced Hessian of the Lagrangian function is used with Variable reduction method which preserves the structure of the Jacobian in representing the null space basis matrix. By updating the reduced Hessians only of which the dimension is the number of design variables, the storage requirement for Hessians is reduced by a large amount. In addition, the flow part of the Jacobian can be computed analytically. The algorithm with a line search globalization is described. A global and local analysis is provided with a modification of the paper by Byrd and Nocedal [Mathematical Programming 49(1991) pp 285-323] in which they analyzed the similar algorithm with the Orthogonal factorization method which assumes the orthogonality of the null space basis matrix. Numerical results are obtained and compared favorably with results from the Black box method - unconstrained optimization formulation. / Ph. D.

This dissertation is concerned with the global optimization of polynomial programming problems, and a detailed treatment of its particular special case, the indefinite quadratic programming problem. Polynomial programming problems are generally nonconvex, involving the optimization of a polynomial objective function over a compact feasible region defined in terms of polynomial constraints. These problems arise in a variety of applications in the context of engineering design and chemical process design situations. Despite the wide applicability of these classes of problems, there is a significant gap between the theory and practice in this field, principally due to the challenge faced in solving these rather difficult problems. The purpose of this dissertation is to introduce new solution strategies that assist in closing this gap. For solving polynomial programming problems, we present a branch and bound algorithm that uses a Reformulation Linearization Technique (RLT) to generate tight linear programming relaxations. This bounding scheme involves an automatic reformulation of the problem via the addition of certain nonlinear implied constraints that are generated by using the products of the simple bounding restrictions and, optionally, products involving the structural constraints. Subsequently, in a linearization phase, each distinct nonlinear term in the resulting problem is replaced by a new variable to obtain a linear program. The underlying purpose of this procedure is to closely approximate the convex envelope of the objective function over the convex hull of the feasible region. Various implementation issues regarding the derivation of such a bounding problem using the RLT, and the dominance of such bounds over existing alternative schemes, are investigated, both the- theoretically and computationally. The principal thrust of the proposed method is to construct a tight linear programming relaxation of the problem via an appropriate RLT procedure, and to use this in concert with a suitable partitioning strategy, in order to derive a practically effective algorithm that is theoretically guaranteed to be globally convergent. To address various implementation issues, empirical experiments are conducted using several test problems from the literature, many motivated by the aforementioned practical applications. Our results on solving several such practical engineering design problems demonstrate the viability of the proposed approach. This approach is also specialized and further enhanced for the particular class of indefinite (and concave) quadratic programming problems. These problems are important in their own right, and arise in many applications such as in modelling economies of scale in a cost structure, in location-allocation problems, several production planning and risk management problems, and in various other mathematical models such as the maximum clique problem and the jointly constrained bilinear programming problem. The proposed algorithm is more than just a specialization of the polynomial programming approach; it involves new, nontrivial extensions that exploit the particular special structure of the problem, along with many additional supporting features that improve the computational efficiency of the procedure. Certain types of nonlinearities are also retained and handled implicitly within the bounding problem to obtain sharper bounds. Computational results are presented on a set of test problems from the literature to demonstrate the efficiency of the approach. (One of these test problems had not previously been solved to optimality.) It is shown that for many problems, including randomly generated problems having up to 50 variables, the initial relaxation itself produces an optimal or a near optimal solution. This is significant in that the proposed methodology affords an approach whereby such hard nonconvex problems can be practically solved via a single or a few higher dimensional linear programming problems. / Ph. D.

Class numbers of algebraic number fields are central invariants. Once the underlying field has an infinite unit group they behave very irregularly due to a non-trivial regulator. This phenomenon occurs already in the simplest case of real quadratic number fields of which very little is known. Hooley derived a conjectural formula for the average of class numbers of real quadratic fields. In this thesis we extend his methods to obtain conjectural formulae and bounds for any moment, i.e., the average of an arbitrary real power of class numbers. Our formulae and bounds are based on similar (quite reasonable) assumptions of Hooley's work. In the final chapter we consider the case of the -1 power from a numerical point of view and develop an efficient algorithm to compute the average for the -1 class number power without computing class numbers.

Class numbers of algebraic number fields are central invariants. Once the underlying field has an infinite unit group they behave very irregularly due to a non-trivial regulator. This phenomenon occurs already in the simplest case of real quadratic number fields of which very little is known. Hooley derived a conjectural formula for the average of class numbers of real quadratic fields. In this thesis we extend his methods to obtain conjectural formulae and bounds for any moment, i.e., the average of an arbitrary real power of class numbers. Our formulae and bounds are based on similar (quite reasonable) assumptions of Hooley's work. In the final chapter we consider the case of the -1 power from a numerical point of view and develop an efficient algorithm to compute the average for the -1 class number power without computing class numbers.

In recent years, the moments of L-functions has been a topic of growing interest in the field of analytic number theory. New techniques, including applications of Random Matrix Theory and multiple Dirichlet series, have lead to many well-posed theorems and conjectures for the moments of various L-functions. In this thesis, we theoretically and numerically examine the integral moments of quadratic Dirichlet $L$-functions. In particular, we exhibit and discuss the conjectures for the moments which result from the applications of Random Matrix Theory, number theoretic heuristics, and the theory of multiple Dirichlet series. In the case of the cubic moment, we further numerically investigate the possible existence of additional lower order main terms.