Teaching Quadratic Problem Solving

We consider the analytic center based column generation algorithm for the problem of finding a feasible point in a set defined by a finite number of convex quadratic inequalities. At each iteration the algorithm computes an approximate analytic center of the set defined by the intersection of quadratic inequalities generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise a quadratic inequality violated at the current center is selected and a new quadratic cut (defined by a convex quadratic inequality) is placednear the approximate center. As the number of cuts increases, the set defined by their intersection shrinks and the algorithm eventually finds a solution to the problem. Previously, similar analytic center based column generation algorithms were studied first for the linear feasibility problem and later for the general convex feasibility problem. Our method differs from these early methods in that we use "quadratic cuts" in the computation instead of linear cuts. Moreover, our method has a polynomial worst case complexity of $O(n\ln \frac{1}{\varepsilon})$ on the total number of cuts to be used, where n is the number of convex quadratic polynomial inequalities in the problem and $\varepsilon$ is the radius of the largest ball contained in the feasible set. In contrast, the early column generation methods using linear cuts can only solve the convex quadratic feasibility problem in pseudopolynomial time.

In this paper, we consider the Optimal Power Flow (OPF) problem which consists in determining the power production at each bus of an electric network by minimizing the production cost. Our contribution is an exact solution algorithm for the OPF problem. It consists in a spatial branch-and-bound algorithm based on a compact quadratically-constrained convex relaxation. It is computed by solving the semidefinite rank relaxation of OPF once at the root node of the algorithm. An important result is that the optimal value of our compact relaxation is equal to the rank relaxation value. Then, at every sub-nodes of our branch-and-bound, the lower bound is obtained by solving a quadratic convex problem instead of an SDP. Another contribution is that we add only O(n+m) variables that model the squares of the initial variables, where n is the number of buses in the power system and m the number of transmission lines, to construct our relaxation. Then, since the relations between the initial and auxiliary variables are non-convex, we relax them to get a quadratic convex relaxation. Finally, in our branch-and-bound algorithm, we only have to force a reduced number of equalities to prove global optimality. This quadratic convex relaxation approach is here tailored to the OPF problem, but it can address any application whose formulation is a quadratic optimization problem subject to quadratic equalities and ring constraints. Our first experiments on instances of the OPF problem show that our new algorithm Compact OPF (COPF) is more efficient than the standard solvers and other quadratic convex relaxation based methods we compare it with.

In several applications in science and engineering, different types of matrix problems emerge from the discretization of partial differential equations. This thesis is devoted to the development of new algorithms to solve this kind of problems. In particular, when the matrices involved are sparse and non-symmetric. The new algorithms are based on the Induced Dimension Reduction method [IDR(s)]. IDR(s) is a Krylov subspace method originally proposed in 2008 to solve systems of linear equations. IDR(s) has received considerable attention due to its stable and fast convergence. It is, therefore, natural to ask if it is possible to extend IDR(s) to solve other matrix problems, and if so, to compare those extensions with other well-established methods. This work aims to answer these questions. The main matrix problems considered in this dissertation are: the standard eigenvalue problem, the quadratic eigenvalue problem, the solution of systems of linear equations, the solution of sequences of systems of linear equations, and linear matrix equations. We focus on examples that arise from the discretization of partial differential equations.

We prove a sufficient global optimality condition for the problem of minimizing a quadratic function subject to quadratic equality constraints where the variables are allowed to take values −1 and 1. We extend the condition to quadratic problems with matrix variables and orthonormality constraints, and in particular to the quadratic assignment problem.

In this paper, we use the solutions of forward-backward stochastic differential equations with random jumps to get the optimal controls for forward-backward stochastic linear quadratic optimal control problem and the Nash equilibrium point for nonzero sum differential games problem.

This paper deals with the problem of estimating the steering direction of a signal, embedded in Gaussian disturbance, under a general quadratic inequality constraint, representing the uncertainty region of the steering. We resort to the maximum likelihood (ML) criterion and focus on two scenarios. The former assumes that the complex amplitude of the useful signal component fluctuates from snapshot to snapshot. The latter supposes that the useful signal keeps a constant amplitude within all the snapshots. We prove that the ML criterion leads in both cases to a fractional quadratically constrained quadratic problem (QCQP). In order to solve it, we first relax the problem into a constrained fractional semidefinite programming (SDP) problem which is shown equivalent, via the Charnes-Cooper transformation, to an SDP problem. Then, exploiting a suitable rank-one decomposition, we show that the SDP relaxation is tight and give a procedure to construct (in polynomial time) an optimal solution of the original problem from an optimal solution of the fractional SDP. We also assess the quality of the derived estimator through a comparison between its performance and the constrained Cramer Rao lower Bound (CRB). Finally, we give two applications of the proposed theoretical framework in the context of radar detection.

We consider quadratic eigenvalue problems with large and sparse matrices depending on a parameter. Problems of this type occur, for example, in the stability analysis of spatially discretized and p...

The purpose of the current study is to report a part of our larger project whose focus is to understand a relationship between students' units coordination and K-12 school mathematics.In particular, in this paper we report how students who exhibit distinct levels of units coordinations used their knowledge of proportion to solve quadratic function problems of the form y ax 2 =. To this end, three 7th grade students all of whom assimiliated whole number problem situations with three levels of units but showed different levels for fraction problems were chosen.We carried out clinical interviews not only to understand their ability to coordinate units but to understand their problem solving process of proportion and the quadratic function problems.The analysis suggest that their abilities to coordinate units influenced their ways to solving proportion problems, and in turn influenced their ways to solve the specific form of quadratic functions.We have finalized our study by discussing how students' ability to construct and coordinate units, their proportion knowledge, and their knowledge associated with expressing the specific type of quadractic functions could be related.