Linear inequality constraints

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

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... 

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. 

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. 

Example 1. The 2-hot/2-cold streams example studied by [10], with problem data presented in Table 1, is illustrated. With these parameters, the multi-objective MILP formulation has 408 linear equality constraints, 760 linear inequality constraints, 12 binary variables, and 545 positive continuous variables. Notably, the restriction of MEUmax = 6 in Eq. (7) will be removed should the minimum number of matches be simultaneously taken into account as one of the design objectives. 

First, one can experimentally measure as many fluxes (or more) as the dimension of the null-space, so as to uniquely calculate the remaining fluxesJ " This approach is called metabolic flux analysis. Alternatively, an objective of the metabolic network can be chosen to computationally explore the best use of the metabolic network by a given metabolic genotype. Herein, we pursue the second option. The solution to Eq. (7) subject to the linear inequality constraints can be formulated as a linear programming (LP) problem, in which one finds fhe flux distribution that minimizes a particular objective. Mathematically, the LP problem is stated as ... 

Figure 18.1 Linear objective function (a) unconstrained (b) subject to linear inequality constraint, x Sx.

The elements of matrix of linear inequality constraints Cj, are composed by coefficients of incoming edges to final nodes F. And the coefficients values of matrix of linear inequality constraints are equal to 4. The elements of low flow boundary vector of pipeline edges are composed of 0. The values of elements of maximum flow boundary vector are dependent on pipeline diameter, pipeline length, the pressure in the system, specific gravity of the gas and other coefficients. 

The MFC set points are calculated so that they maximize or minimize an economic objective function. The calculations are usually based on linear steady-state models and a simple objective function, typically a linear or quadratic function of the MVs and CVs. The linear model can be a linearized version of a complex nonlinear model or the steady-state version of the dynamic model that is used in the control calculations. Linear inequality constraints for the MVs and CVs are also included in the steady-state optimization. The set-point calculations are repeated at each sampling instant because the active constraints can change frequently due to disturbances, instrumentation, equipment availability, or varying process conditions. 

Because the set-point calculations are repeated as often as every minute, the steady-state optimization problem must be solved quickly and reliably. If the optimization problem is based on a linear process model, linear inequality constraints, and either a linear or a quadratic cost function, the hnear and quadratic programming techniques discussed in Chapter 19 can be employed. 

The amplitude of a radiated seismic wave contains far more information about the earthquake mechanism than does its polarity alone, so amplitude data can be valuable in studies of non-DC earthquakes. Moreover, because seismic-wave amplitudes are linear functions of the moment-tensor components, determining moment tensors from observed amplitudes is a linear inverse problem, which can be solved by standard methods such as least squares. Conventional least-squares methods, however, cannot invert polarity observations such as first motions, which typically are the most abundant data available. Linear programming methods, which can treat linear inequalities, are well suited to inverting observations that include both amplitudes and polarities (Julian 1986). In this approach, bounds are placed on observed amplitudes, so that they can be expressed as linear inequality constraints. Polarities are already in... 

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