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Quadratic Programming Theory
Published in Michael J. Best, Quadratic Programming with Computer Programs, 2017
Numerical methods for linear programming are based on the fact that the feasible region possesses only a finite number of extreme points and at least one extreme point is optimal. The analogous quantity to an extreme point for quadratic programming is a quasistationary point which is defined as follows. For any x0 ∈ R, define I(x0)={i|a′ix0=bi,1≤i≤m};
Linear Programming
Published in Michael W. Carter, Camille C. Price, Ghaith Rabadi, Operations Research, 2018
Michael W. Carter, Camille C. Price, Ghaith Rabadi
We have seen in our graphical solutions that, if an optimal solution exists, it occurs at an extreme point of the feasible region. This fundamental property of linear programming problems is the foundation for a general solution method called the Simplex method. Because only the finitely many extreme points need be examined (rather than all the points in the feasible region), an optimal solution may be found systematically by considering the objective function values at the extreme points. In fact, in actual practice, only a small subset of the extreme points need be examined. The following sections will demonstrate how the Simplex method is able to locate optimal solutions with such efficiency.
Optimal Power Flow
Published in J.C. Das, Power System Analysis, 2017
We noted in Chapter 15 that, in the simplex method, we go from one extreme point to another extreme point (Example 15.8). Figure 16.8 shows a comparison of the two methods. The simplex method solves an LP problem, starting with one extreme point on the boundary of the feasible region, then going to a better neighboring extreme point along the boundary, and finally stopping at the optimum extreme point. The IP method stays in the interior of the polytope.
MOS Amplifier Design Methodology for Optimum Performance
Published in IETE Journal of Research, 2020
Abir J. Mondal, Paromita Bhattacharjee, Pinaki Chakraborty, Bidyut K. Bhattacharyya
A simple graphical characterization of the constrained optimization problem shows that we can graph the feasible solutions together to form a geometric object. The geometric object may take a shape having at sides as well as curved sides or a shape of a polytope, which has only at sides and can exist in any general number of dimensions. The number of sides thus formed depends on the number of circuit constraint equations subjected to the objective function. Each constraint imposes a half-plane. In order to determine these half-planes, constraint equations need to be plotted by replacing the inequality with equality and then verifying whether or not some specific point that does not satisfy the equality satisfies the inequality constraint. The feasible region is the intersection of these half-planes [11]. This feasible region can be regarded as the accessible design space. For linear programming, the optimal solution typically exists at one of the intersection points. The simplex algorithm implements this by surveying the extreme points. However, for a NLP problem, it is not necessary for the optimal solution to be present at an extreme point. In NLP, an optimal solution might exist in the interior of the design space as well. Interior point-based methods have been developed to solve such NLP problems. Interior point-based methods in its very basic form obtain the best solution by traversing the interior of the feasible region. Research on interior point methods traces back to the work of Fiacco and McCormick in 1968. Attention to these methods grew after the work of Karmarkar in 1984 [11].
Optimizing Nuclear Material Accounting and Measurement Systems
Published in Nuclear Technology, 2018
Nicolas Shugart, Benjamin Johnson, Jeffrey King, Alexandra Newman
Integer programming is a branch of optimization that restricts at least some of the decision variables in the problem to integer values.14 The primary benefit of integer programming relative to other optimization methods is the ability to model binary decisions, represented by a value of one if a decision is to be made and zero if a decision is not to be made. Binary variables also allow for logical constructs such as if-then statements. Mixed-integer programs include continuous, in addition to integer, variables. Algorithms for integer programs enumerate a subset of feasible, nondominated solutions to determine an optimum.14 The limitation of integer programs is that they are nonconvex, which makes them more difficult to solve than their convex, linear counterparts. Linear programming algorithms can search extreme point solutions, i.e., vertices in a convex set, for a local optimum, which is guaranteed to be the global optimum due to convexity. A drawback of linear programs is that they contain only linear constraints and continuous variables, which limits their applicability to many problems.14
Modelling uncertainty with generalized credal sets: application to conjunction and decision
Published in International Journal of General Systems, 2018
Andrey G. Bronevich, Igor N. Rozenberg
Let us remind that P is an extreme point of a convex set if the equation has no solution for and such that and . If the set has a finite number of extreme points , then any can be represented as a convex sum of extreme points, i.e. , where and .