Constraint matrix

The staircase matrix structure of the 2S-MILP (see Figure 9.9) is exploited by 2S-MILP-specific decomposition based algorithms [9,10], The constraint matrix of the 2S-MILP consists of 12 subproblems Wm that are tied together by the first-stage variables x and the corresponding matrix column [ATj. .. Th]r. The main steps of decomposition based algorithms for 2S-MILPs are ... 

In the rest of this chapter, we assume that the rows of the constraint matrix A are linearly independent, that is, rank (A) = m. If a slack variable is inserted in every row, then A contains a submatrix that is the identity matrix. In the preceding example, if we insert a slack variable x5 into the equality ... 

Modem LP solvers can solve very large LPs very quickly and reliably on a PC or workstation. LP size is measured by several parameters (1) the number of variables n, (2) the number of constraints m, and (3) the number of nonzero entries nz in the constraint matrix A. The best measure is the number of nonzero elements nz because it directly determines the required storage and has a greater effect on computation time than n or m. For almost all LPs encountered in practice, nz is much less than mn, because each constraint involves only a few of the variables jc. The problem density 100(nz/mn) is usually less than 1%, and it almost always decreases as m and n increase. Problems with small densities are called sparse, and real world LPs are always sparse. Roughly speaking, a problem with under 1000 nonzeros is small, between 1000 and 50,000 is medium-size, and over 50,000 is large. A small problem probably has m and n in the hundreds, a medium-size problem in the low to mid thousands, and a large problem above 10,000. 

Consider the process flowsheet shown in Figure El6.4, which was used by Rollins and Davis (1993) in investigations of gross error detection. The seven stream numbers are identified in Figure El6.4. The overall material balance can be expressed using the constraint matrix Ay = 0, where A is given by... 

A 1...NE,1...NV) CONSTRAINT MATRIX COEFFICIENTS A(l...NE.NV+l CONSTRAINT RIGHT HAND SIDES OBJECTIVE FUNCTION COEFFICIENTS... 

Afh (q) is the current element of a matrix A (q) called the constraint-matrix. A (q) is a real symmetric matrix whose diagonal elements are positive it depends on the generalized coordinates as variables and parametrically on the constraints. This matrix possesses an inverse since det A(q) = 0 is not possible it would correspond to a supplementary relationship between the coordinates only, i.e a supplementary holonomic constraint. 

On the basis of Eqs. (40) and (42), one may state that — once the constraint matrix A (q) is known - using the hamiltonian formalism to study the dynamics of a constrained system amounts to... 

An alternative to the method of Lagrange multipliers for imposing the necessary constraints is sketched below. It derives a lower dimensional unconstrained problem from the original constrained problem by using an orthogonal basis for the null space of the constraint matrix. This method is well suited to the potentially rank-deficient problem at hand, where steps may be taken to... 

This is a set packing formulation and is NP-hard [89]. There are special cases under which the structure of this problem simplifies and allows for polynomial time solutions. Many special cases arise out of constraints that reduce the constraint matrix to be totally unimodular [35]. A common example is the case where adjacent plots of land are being sold and bidders might want multiple plots but they need to be adjacent. However, real world problems will often not satisfy the fairly severe restrictions that provide a totally unimodular constraint matrix. Moreover, if the bidding language is not expressive then this can interact with the incentive properties of an auction because a bidder is not able to express her true valuation, even if that would be her equilibrium strategy. We wait until Section 4 for an extensive discussion of the interaction between computational constraints and incentives. 

To describe the dual to SPP let 1 denote the m-vector of all 1 s and the column of the constraint matrix A. The (superadditive) dual to SPP is the problem of finding a superadditive, non-decreasing function F that solves... 

In simulations of dihedral rotations, atoms are moved independently and bond-lengths and angles can be fixed at their equilibrium values by inverting the constraint matrix directly or... 

The constraint matrix T depicts the effect of the constraining matrix on the inclusion and is a function of matrix material properties and ellipsoidal inclusion shape. It represents the piezoelectric analog to Eshelby s tensor in the elastic case, see Dunn and Taya [66]. Expressions for cylindrical inclusions to model fibrous composites are provided by Dunn and Taya [67] (this reference uses a different notation, T is called S). Equating Eqs. (5.12) and (5.13) and making use of Eq. (5.14) to replace the eigenfields Z and Eq. (5.11) to eliminate the perturbation fields Z after some manipulations, leads to... 

Herein, G is the n x rip constraint matrix, K,D M are the stiffness-, damping-and mass matrices, respectively, and z are time dependent kinematical excitations. To keep formulas short we assume u i) = 0. Furthermore, we want to exclude redundant constraints by assuming rank(G) = n. ... 

Lemma 2.6.5 The matrix pencil fiE — A of the linear mechanical system (2.6.2) is regular if and only if the n x Up constraint matrix G has full rank. 

C is constraints matrix and A is transformation matrix. Detailed way of finding solution to this equation is presented in (Fossen, 2010). 

---