Disentangling Orthogonal Matrices

Motivated by a certain molecular reconstruction methodology in cryo-electron microscopy, we consider the problem of solving a linear system with two unknown orthogonal matrices, which is a generalization of the well-known orthogonal Procrustes problem. We propose an algorithm based on a semi-definite programming (SDP) relaxation, and give a theoretical guarantee for its performance. Both theoretically and empirically, the proposed algorithm performs better than the na\"{i}ve approach of solving the linear system directly without the orthogonal constraints. We also consider the generalization to linear systems with more than two unknown orthogonal matrices

Particulars of Non-Linear Optimization

We are providing a concise introduction to some methods for solving non-linear optimization problems. In mathematics,non-linear programming (NLP) is the process of solving an optimization problem defined by a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are non-linear. It is the sub-field of mathematical optimization that deals with problems that are not linear. This dissertation conducts its study on the theory that are necessary for understanding and implementing the optimization and an investigation of the algorithms such as Wolfe's Algorithm, Dinkelbach's Algorithm and etc. are available for solving a special class of the non-linear programming problem, quadratic programming problem which is included in the course of study. Optimization problems arise continuously in a wide range of fields such as Power System Control and thus create the need for effective methods of solving them. We discuss the fundamental theory necessary for the understanding of optimization problems, with particular programming problems and the algorithms that solve such problems

A Method for Solving Linear Programming Problems with Unknown Parameters

A method is proposed for solving a problem of linear programming with unknown constraints. The form of the unknown constraints needs to be identified by a proper choice of the observation data. The present method is based upon a bicriterion formulation to the joint identification and optimization problem. A parametric approach is used to obtain an efficient solution to the bicriterion problem. Further, a decomposition into subproblems easily solvable is introduced. The interaction between subproblems is coordinated by an adjustment of a scalar parameter varying over the unit interval

A Global Approach for Solving Edge-Matching Puzzles

We consider apictorial edge-matching puzzles, in which the goal is to arrange a collection of puzzle pieces with colored edges so that the colors match along the edges of adjacent pieces. We devise an algebraic representation for this problem and provide conditions under which it exactly characterizes a puzzle. Using the new representation, we recast the combinatorial, discrete problem of solving puzzles as a global, polynomial system of equations with continuous variables. We further propose new algorithms for generating approximate solutions to the continuous problem by solving a sequence of convex relaxations

Efficient Solving of Quantified Inequality Constraints over the Real Numbers

Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are $\leq$ and $<$. Solving such constraints is an undecidable problem when allowing function symbols such $\sin$ or $\cos$. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques