Quadratic Programming

Abstract: This paper proves that non-convex quadratically constrained quadratic programs have an exact semidefinite relaxation when their underlying graph is acyclic, provided the constraint set satisfies a certain technical condition. When the condition is not satisfied, we propose a heuristic to obtain a feasible point starting from a solution of the relaxed problem. These methods are then demonstrated to provide exact solutions to a richer class of optimal power flow problems than previously solved.

A simulation-based interval quadratic waste load allocation (IQWLA) model was developed for supporting river water quality management. A multi-segment simulation model was developed to generate water-quality transformation matrices and vectors under steady-state river flow conditions. The established matrices and vectors were then used to establish the water-quality constraints that were included in a water quality management model. Uncertainties associated with water quality parameters, cost functions, and environmental guidelines were described as intervals. The cost functions of wastewater treatment units were expressed in quadratic forms. A water-quality planning problem in the Changsha section of Xiangjiang River in China was used as a study case to demonstrate applicability of the proposed method. The study results demonstrated that IQWLA model could effectively communicate the interval-format uncertainties into optimization process, and generate inexact solutions that contain a spectrum of potential wastewater treatment options. Decision alternatives can be generated by adjusting different combinations of the decision variables within their solution intervals. The results are valuable for supporting local decision makers in generating cost-effective water quality management strategies.

This paper deals with the control laws
reconguration of nonlinear systems, by using a
Fuzzy-Model-based Predictive Control (FMPC). It
should be noted that the studied systems are writ-
ten in the quasi-linear parametric varying (quasi-
LPV) form. This FMPC strategy is developed to
preserve closed-loop stability in the nominal and
actuator faulty case. Fault accommodation by per-
turbations rejection is presented. This step is done
by interpolation-based control to cover the entire
area of operation. To allow the process to maintain
current performances closed to desired performances,
a dynamic optimizer is used. Our contribution comes
from the combination of several aspects: fuzzy model,
quadratic programming and faults decoupling princi-
ple. The linearization around a family of equilibrium
points is also studied. The operating points are appro-
priately congured by a set of variables called premise.

In this paper we introduce the class of quadratically optimal (bi-matrix) games, which are bi-matrix games whose set of equilibrium points contain all pairs of probability vectors which maximize the expected pay-off of some pay-off matrix. We call the equilibrium points obtained in this way, quadratically optimal equilibrium points. We prove the existence of quadratically optimal equilibrium points of identical bi-matrix games, i.e. bi-matrix games for which the two pay-off matrices are equal, from which it easily follows that weakly potential bi-matrix games (a generalization of potential bi-matrix games) are quadratically optimal. We also show that those weakly potential square bi-matrix games which have potential matrices that are two-way matrices are quadratically optimal symmetric games, i.e. there exists a square pay-off matrix whose expected pay-off maximizing probability vectors subject to the two probability vectors (row probability vector and column probability vector) being equal, are equilibrium points of the bi-matrix game. None of our results require using a fixed-point theorem argument in the proofs. We show by means of an example of a 22 identical symmetric potential bi-matrix game that for every potential matrix of the game, the set of pairs of probability distributions that maximizes the expected pay-off of the potential matrix is a strict subset of the set of equilibrium points of the potential game.

We investigate the application of Support Vector Machines (SVMs) in computer vision. SVM is a learning technique developed by V. Vapnik and his team (AT&T Bell Labs., 1985) that can be seen as a new method for training polynomial, neural network, or Radial Basis Functions classifiers. The decision surfaces are found by solving a linearly constrained quadratic programming problem. This optimization problem is challenging because the quadratic form is completely dense and the memory requirements grow with the square of the number of data points. We present a decomposition algorithm that guarantees global optimality, and can be used to train SVM's over very large data sets. The main idea behind the decomposition is the iterative solution of sub-problems and the evaluation of optimality conditions which are used both to generate improved iterative values, and also establish the stopping criteria for the algorithm. We present experimental results of our implementation of SVM, and demonstrate the feasibility of our approach on a face detection problem that involves a data set of 50,000 data points

A support vector machine (SVM) learns the decision surface from two different classes of the input points. In several applications, some of the input points are misclassified and each is not fully allocated to either of these two groups. In this paper a bi-objective quadratic programming model with fuzzy parameters is utilized and different feature quality measures are optimized simultaneously. An α-cut is defined to transform the fuzzy model to a family of classical bi-objective quadratic programming problems. The weighting method is used to optimize each of these problems. For the proposed fuzzy bi-objective quadratic programming model, a major contribution will be added by obtaining different effective support vectors due to changes in weighting values. The experimental results, show the effectiveness of the α-cut with the weighting parameters on reducing the misclassification between two classes of the input points. An interactive procedure will be added to identify the best com...

A support vector machine (SVM) learns the decision surface from two different classes of the input points. In several applications, some of the input points are misclassified and each is not fully allocated to either of these two groups. In this paper a bi-objective quadratic programming model with fuzzy parameters is utilized and different feature quality measures are optimized simultaneously. An α-cut is defined to transform the fuzzy model to a family of classical bi-objective quadratic programming problems. The weighting method is used to optimize each of these problems. For the proposed fuzzy bi-objective quadratic programming model, a major contribution will be added by obtaining different effective support vectors due to changes in weighting values. The experimental results, show the effectiveness of the α-cut with the weighting parameters on reducing the misclassification between two classes of the input points. An interactive procedure will be added to identify the best compromise solution from the generated efficient solutions. The main contribution of this paper includes constructing a utility function for measuring the degree of infection with coronavirus disease (COVID-19).

This paper presents three-level quadratic programming problem with random rough coefficient in constraints. At the
first phase of the solution algorithm and to avoid the complexity of this problem, we begin with converting the rough
nature in constraints into equivalent crisp using intervals technique. At the second phase, a membership function is
constructed to develop a fuzzy model for obtaining a compromise solution of the three-level quadratic programming
problem. In addition, the theoretical results are illustrated with the help of a numerical example.

The modeling and control problem for a grid-connected photovoltaic (PV) power electronic system, which includes a dc/dc boost converter, an inverter and a filter are considered. A linear complementarity (LC) dynamic model of the PV system allows the design of a model predictive controller (MPC). Dynamic models of the subsystems are obtained and merged in order to represent the whole PV system in a compact and comprehensive LC model, which is valid for all operating modes of power converters and PV cells involved in the energy conversion process. A finite-control-set MPC problem is formulated as a mixed-integer quadratic program subject to the dynamic LC model and pulse width modulators. The minimization of an objective function aimed at tracking dc voltage and grid current references provides directly the commands for the switches of the boost converter and inverter. Numerical results show the effectiveness of the proposed strategy for maximum power point tracking and synchronization to the grid under dynamic scenarios characterized by variations of the solar irradiance.

This paper presents quadratic integer programming as a modeling technique to formulate the analytical models for financial markets. Financial market analysis has been a promising area of research for the last three decades. The proposed models aim to cater for several scenarios for analysis and prediction of future trends in capital, gold and currency markets. These models after translation into hyper graphs would be solved through clustering (data mining). The implementation of this research would be the basis of a decision support system. It may prove to be an analysis tool to evaluate the economic status of a country statistically in a more accurate manner

Particle swarm optimization is used in several combinatorial optimization problems. In this work, particle swarms are used to solve quadratic programming problems with quadratic constraints. The approach of particle swarms is an example for interior point methods in optimization as an iterative technique. This approach is novel and deals with classification problems without the use of a traditional classifier. Our method determines the optimal hyperplane or classification boundary for a data set. In a binary classification problem, we constrain each class as a cluster, which is enclosed by an ellipsoid. The estimation of the optimal hyperplane between the two clusters is posed as a quadratically constrained quadratic problem. The optimization problem is solved in distributed format using modified particle swarms. Our method has the advantage of using the direction towards optimal solution rather than searching the entire feasible region. Our results on the Iris, Pima, Wine, and Thyroid datasets show that the proposed method works better than a neural network and the performance is close to that of SVM.