Linear inequality

Now z — cy < 0 must hold for all j in order to have obtained a solution x° whose components are given by the coefficients expressing P0 as a linear combination of Pi and P2. To impose the condition zy — cy < 0 on the parameter t, is to solve a set of simultaneous—not necessarily linear—inequalities in. Then Pi and P2 would be an optimal basis for this interval of values of. By fixing a value of immediately outside the interval and in the neighborhood of a boundary point, the vector to be eliminated and that to be introduced into the basis are produced in the usual manner, and the process is then repeated. If no value of t satisfies the set of inequalities, then by fixing at a given 0, the usual procedure is used to eliminate a vector and introduce another into the basis. 

P.J. Gemperline, Target transformation factor analysis with linear inequality constraints applied to spectroscopic-chromatographic data. Anal. Chem., 58 (1986) 2656-2663. 

Another method for solving nonlinear programming problems is based on quadratic programming (QP)1. Quadratic programming is an optimization procedure that minimizes a quadratic objective function subject to linear inequality or equality (or both types) of constraints. For example, a quadratic function of two variables x and X2 would be of the general form ... 

Binary terms can be relaxed by using the McCormick underestimator e.g., the binary term xz is replaced by a new variable w and linear inequality constraints... 

The constraints of a two-stage stochastic linear program can be classified into constraints on the first-stage variables only (9.3.2) and constraints on the first and on the second-stage variables (9.3.3). The latter represent the interdependency of the stages. All constraints are represented as linear inequalities with the matrices Aer>x">, Tb6 R 2xn Wm Rm2x 2, and the vectors b Rm2 and K, e R 2. 

As mentioned in Chapter 1, the occurrence of linear inequality constraints in industrial processes is quite common. Inequality constraints do not affect the count of the degrees of freedom unless they become active constraints. Examples of such constraints follow ... 

EXAMPLE 2.9 FORMULATION OF A LINEAR INEQUALITY CONSTRAINT FOR BLENDING... 

Solution. See Figure E4.9 for the region delineated by the inequality constraints. By visual inspection, the region is convex. This set of linear inequality constraints forms a convex region because all the constraints are concave. In this case the convex region is closed. 

Diagram of region defined by linear inequality constraints. 

The feasible region lies within the unshaded area of Figure 7.1 defined by the intersections of the half spaces satisfying the linear inequalities. The numbered points are called extreme points, comer points, or vertices of this set. If the constraints are linear, only a finite number of vertices exist. 

Circular objective function contours and linear inequality constraint. 

In summary, the problem consists of 34 bounded variables (both upper bound and lower bounds) associated with the process, 12 linear equality constraints, 18 nonlinear equality constraints, and 3 linear inequality constraints. 

LINEAR INEQUALITIES FOR DIAGONAL ELEMENTS OF DENSITY MATRICES... 

Necessary and Sufficient Conditions for V-Representability Linear Inequalities from the Orbital Representation... 

G. Constraints on Off-Diagonal Elements from Other Positive-Definite Hamiltonians Linear Inequalities from the Spatial Representation Linking the Orbital and Spatial Representations... 

By imposing enough linear inequalities of this type, one constmcts a polyhedral hull that bounds the set of A-representable Q-matrices. 

The preceding is a rather comprehensive—but not exhaustive— review of N-representability constraints for diagonal elements of reduced density matrices. The most general and most powerful V-representability conditions seem to take the form of linear inequalities, wherein one states that the expectation value of some positive semidefinite linear Hermitian operator is greater than or equal to zero, Tr [PnTn] > 0. If Pn depends only on 2-body operators, then it can be reduced into a g-electron reduced operator, Pq, and Tr[Pg vrg] > 0 provides a constraint for the V-representability of the g-electron reduced density matrix, or 2-matrix. Requiring that Tr[Pg Arrg] > 0 for every 2-body positive semidefinite linear operator is necessary and sufficient for the V-representability of the 2-matrix [22]. 

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