On Degeneracy Issues in Multi-parametric Programming and Critical Region Exploration based Distributed Optimization in Smart Grid Operations

Improving renewable energy resource utilization efficiency is crucial to reducing carbon emissions, and multi-parametric programming has provided a systematic perspective in conducting analysis and optimization toward this goal in smart grid operations. This paper focuses on two aspects of interest related to multi-parametric linear/quadratic programming (mpLP/QP). First, we study degeneracy issues of mpLP/QP. A novel approach to deal with degeneracies is proposed to find all critical regions containing the given parameter. Our method leverages properties of the multi-parametric linear complementary problem, vertex searching technique, and complementary basis enumeration. Second, an improved critical region exploration (CRE) method to solve distributed LP/QP is proposed under a general mpLP/QP-based formulation. The improved CRE incorporates the proposed approach to handle degeneracies. A cutting plane update and an adaptive stepsize scheme are also integrated to accelerate convergence under different problem settings. The computational efficiency is verified on multi-area tie-line scheduling problems with various testing benchmarks and initial states

Solution Techniques for Classes of Biobjective and Parametric Programs

Mathematical optimization, or mathematical programming, has been studied for several decades. Researchers are constantly searching for optimization techniques which allow one to de-termine the ideal course of action in extremely complex situations. This line of scientific inquiry motivates the primary focus of this dissertation — nontraditional optimization problems having either multiple objective functions or parametric input. Utilizing multiple objective functions al-lows one to account for the fact that the decision process in many real-life problems in engineering, business, and management is often driven by several conflicting criteria such as cost, performance, reliability, safety, and productivity. Additionally, incorporating parametric input allows one to ac-count for uncertainty in models’ data, which can arise for a number of reasons, including a changing availability of resources, estimation or measurement errors, or implementation errors caused by stor-ing data in a fixed precision format. However, when a decision problem has either parametric input or multiple objectives, one cannot hope to find a single, satisfactory solution. Thus, in this work we develop techniques which can be used to determine sets of desirable solutions. The two main problems we consider in this work are the biobjective mixed integer linear program (BOMILP) and the multiparametric linear complementarity problem (mpLCP). BOMILPs are optimization problems in which two linear objectives are optimized over a polyhedron while restricting some of the decision variables to be integer. We present a new data structure in the form of a modified binary tree that can be used to store the solution set of BOMILP. Empirical evidence is provided showing that this structure is able to store these solution sets more efficiently than other data structures that are typically used for this purpose. We also develop a branch-and-bound (BB) procedure that can be used to compute the solution set of BOMILP. Computational experiments are conducted in order to compare the performance of the new BB procedure with current state-of-the-art methods for determining the solution set of BOMILP. The results provide strong evidence of the utility of the proposed BB method. We also present new procedures for solving two variants of the mpLCP. Each of these proce-dures consists of two phases. In the first phase an initial feasible solution to mpLCP which satisfies certain criteria is determined. This contribution alone is significant because the question of how such an initial solution could be generated was previously unanswered. In the second phase the set of fea-sible parameters is partitioned into regions such that the solution of the mpLCP, as a function of the parameters, is invariant over each region. For the first variant of mpLCP, the worst-case complex-ity of the presented procedure matches that of current state-of-the-art methods for nondegenerate problems and is lower than that of current state-of-the-art methods for degenerate problems. Addi-tionally, computational results show that the proposed procedure significantly outperforms current state-of-the-art methods in practice. The second variant of mpLCP we consider was previously un-solved. In order to develop a solution strategy, we first study the structure of the problem in detail. This study relies on the integration of several key concepts from algebraic geometry and topology into the field of operations research. Using these tools we build the theoretical foundation necessary to solve the mpLCP and propose a strategy for doing so. Experimental results indicate that the presented solution method also performs well in practice

Design of of model-based controllers via parametric programming

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An Algorithm for Biobjective Mixed Integer Quadratic Programs

Multiobjective quadratic programs (MOQPs) are appealing since convex quadratic programs have elegant mathematical properties and model important applications. Adding mixed-integer variables extends their applicability while the resulting programs become global optimization problems. Thus, in this work, we develop a branch and bound (BB) algorithm for solving biobjective mixed-integer quadratic programs (BOMIQPs). An algorithm of this type does not exist in the literature. The algorithm relies on five fundamental components of the BB scheme: calculating an initial set of efficient solutions with associated Pareto points, solving node problems, fathoming, branching, and set dominance. Considering the properties of the Pareto set of BOMIQPs, two new fathoming rules are proposed. An extended branching module is suggested to cooperate with the node problem solver. A procedure to make the dominance decision between two Pareto sets with limited information is proposed. This set dominance procedure can eliminate the dominated points and eventually produce the Pareto set of the BOMIQP. Numerical examples are provided. Solving multiobjective quadratic programs (MOQPs) is fundamental to our research. Therefore, we examine the algorithms for this class of problems with different perspectives. The scalarization techniques for (strictly) convex MOPs are reviewed and the available algorithms for computing efficient solutions for MOQPs are discussed. These algorithms are compared with respect to four properties of MOQPs. In addition, methods for solving parametric multiobjective quadratic programs are studied. Computational studies are provided with synthetic instances, and examples in statistics and portfolio optimization. The real-life context reveals the interplay between the scalarizations and provides an additional insight into the obtained parametric solution sets

Multi-parametric Analysis for Mixed Integer Linear Programming: An Application to Transmission Planning and Congestion Control

Enhancing existing transmission lines is a useful tool to combat transmission congestion and guarantee transmission security with increasing demand and boosting the renewable energy source. This study concerns the selection of lines whose capacity should be expanded and by how much from the perspective of independent system operator (ISO) to minimize the system cost with the consideration of transmission line constraints and electricity generation and demand balance conditions, and incorporating ramp-up and startup ramp rates, shutdown ramp rates, ramp-down rate limits and minimum up and minimum down times. For that purpose, we develop the ISO unit commitment and economic dispatch model and show it as a right-hand side uncertainty multiple parametric analysis for the mixed integer linear programming (MILP) problem. We first relax the binary variable to continuous variables and employ the Lagrange method and Karush-Kuhn-Tucker conditions to obtain optimal solutions (optimal decision variables and objective function) and critical regions associated with active and inactive constraints. Further, we extend the traditional branch and bound method for the large-scale MILP problem by determining the upper bound of the problem at each node, then comparing the difference between the upper and lower bounds and reaching the approximate optimal solution within the decision makers' tolerated error range. In additional, the objective function's first derivative on the parameters of each line is used to inform the selection of lines to ease congestion and maximize social welfare. Finally, the amount of capacity upgrade will be chosen by balancing the cost-reduction rate of the objective function on parameters and the cost of the line upgrade. Our findings are supported by numerical simulation and provide transmission line planners with decision-making guidance

Brain microstructure by multi-modal MRI: Is the whole greater than the sum of its parts?

The MRI signal is dependent upon a number of sub-voxel properties of tissue, which makes it potentially able to detect changes occurring at a scale much smaller than the image resolution. This "microstructural imaging" has become one of the main branches of quantitative MRI. Despite the exciting promise of unique insight beyond the resolution of the acquired images, its widespread application is limited by the relatively modest ability of each microstructural imaging technique to distinguish between differing microscopic substrates. This is mainly due to the fact that MRI provides a very indirect measure of the tissue properties in which we are interested. A strategy to overcome this limitation lies in the combination of more than one technique, to exploit the relative contributions of differing physiological and pathological substrates to selected MRI contrasts. This forms the basis of multi-modal MRI, a broad concept that refers to many different ways of effectively combining information from more than one MRI contrast. This paper will review a range of methods that have been proposed to maximise the output of this combination, primarily falling into one of two approaches. The first one relies on data-driven methods, exploiting multivariate analysis tools able to capture overlapping and complementary information. The second approach, which we call "model-driven", aims at combining parameters extracted by existing biophysical or signal models to obtain new parameters, which are believed to be more accurate or more specific than the original ones. This paper will attempt to provide an overview of the advantages and limitations of these two philosophies