Non-negativity constraints

If solution gives negative flows or compositions, the non-negativity constraints can then be added end... 

These equations coupled with the non-negativity constraints form a linear program which can be modeled on LINGO as follows ... 

Linear Programming.28—A linear programming problem as defined in matrix notation requires that a vector x 0 (non-negativity constraints) be found that satisfies the constraints Ax <, b, and maximizes the linear function cx. Here x = (xx, , xn), A = [aiy] (i = 1,- -,m j = 1,- , ), b - (61 - -,bm), and c = (cu- -,c ) is the cost vector. With the original (the primal) problem is associated the dual problem yA > c, y > 0, bij = minimum, where y yx,- , ym)-A duality theorem 29 asserts that if either the primal or the dual has a solution then the values of the objective functions of both problems at the optimum are the same. It is a relatively easy matter to obtain the solution vector of one problem from that of the other. 

Fig. 34.13. Score plot (ti vs 12) of the spectra given in Fig. 34.2 after normalisation. The points A and B are the purest spectra in the data set. The points A and B are the spectra at the boundaries of the non-negativity constraint.

Fig. 34.17. Non-negativity constraints m- and n-Iine) applied on the scores of the normalized...

There are, in fact, a number of solutions to the governing equations, but usually (see Chapter 12) only one with positive mole numbers and concentrations. Fortunately, the latter answer is of interest to all but the most abstract-thinking geochemist. The requirement that the iteration produce positive masses is known in chemical modeling as the non-negativity constraint. 

In an s-dimensional space, s vectors at most can be independent. At equilibrium, a rock made of s elements cannot consist of more than s minerals, which implies that at least p—s of the p mole numbers are zero. In order to find the set of independent vectors that minimize the energy, we first rearrange the order of variables and split the vector n into two parts. The first part is the vector nB made of s base variables, and the second part is the vector F of (p —s) free variables. Provided the base variables are non-negative, the non-negativity constraints can be satisfied by setting the free variables to zero. For the vector n to be a feasible solution, it should also satisfy the recipe equation, i.e.,... 

There are exceptions to the universality of this non-negativity constraint e.g. CD or ESR spectra can be negative. Apart from that, both spectroscopies produce a signal that is a linear function of concentration and thus the equivalent of Beer-Lambert s law holds. In other words, the equation Y=CA applies and thus also the ALS algorithm. 

The total investment expenditures incurred at a site have to be calculated in two steps. Equation (3.10) calculates the investments per plant. These are aggregated to the site level and adjusted for government investment incentives, defined as percentage of total investments, in equation (3.11). Investment expenditures are allocated to the time period preceding the commissioning of the technical capacity. A non-negativity constraint (3.54) ensures that plant/production line shutdowns do not lead to negative investment expenditures. 

Closure costs for existing plants incurred at a site are again determined in two steps. Plant closure costs are calculated in equation (3.14) which, in combination with the non-negativity constraint (3.57), ensures that no negative closure costs are calculated. Closure costs are allocated to the time period during which the capacity is decommissioned. To account for lead times in closure decisions no plant closures can occur in the first period of the planning horizon. Equation (3.15) aggregates the closure costs to the site level. 

The calculation of severance payments in equations (3.16) and (3.17) considers the natural fluctuation rate and personnel requirements as specified in equation (3.49). Again, non-negativity constraint (3.72) ensures that in case of personnel increases no negative severance payments are calculated. The same type of formulation could also be used to model training costs for newly hired employees but this was considered to be of minor relevance in the application cases pursued so far. 

Equation (3.47) calculates the tariffs that have to be paid for import of raw materials. Duty drawbacks are considered by charging tariffs only for raw materials required for production of intermediate (second line) and finished goods (first line) that are not re-exported. The amount of intermediates not re-exported is adjusted for those intermediates that are transformed into goods subsequently re-exported at the same site. The formulation rests on the assumption that only one raw material source is used per site and that if a required intermediate is available locally the local source is used. Prices are converted from the currency used by the raw material supplier to the currency of the consuming site. Same as with the valuation of finished goods for tariff calculation the full transport costs are included in the tariff value of the raw materials and intermediates. Import tariffs for intermediates imported from another site are calculated in equation (3.48). The formulation differs from the one for raw materials because the source of the intermediates is not predetermined and the transfer price does not contain transportation costs. The net volumes of the intermediates which are not re-exported are calculated by subtracting the quantities contained in exported products from the total quantity imported. A non-negativity constraint sets the value to 0 if the respective intermediate is not imported from site s. ... 

Restrictions (3.53) to (3.75) are the common non-negativity constraints. Variable Domains... 

The integration of capital expenditures for the shared resources as shown in equation (3.11a) rests on the assumption that the number of equipments is correlated to the number of production lines. If the model can independently select the shared resource capacity, it is theoretically possible that the number of shared resources operated increases while the number of production lines used decreases. In this case a separate calculation of the investment expenditures for shared resources is required to avoid that "negative" capital expenditures from capacity reductions that are eliminated via the non-negativity constraint (3.54) offset the expenditures for shared resource installations. 

The objective function (4.1) minimizes the efficiency measure E. For the smallest E obtained the slack variables are maximized. This objective hierarchy is achieved by including the "very small" parameter 8 that subordinates the maximization of the slack variables under the minimization of E. Equations (4.2) and (4.3) specify the output and input factor comparisons. The slack variables contain the surplus of output factors and underconsumption of input factors respectively as compared to the virtual DMU. The weight parameters 7ru are determined by the optimization model and describe the linear combination of real DMUs constituting the virtual DMU. Restriction (4.4) contains non-negativity constraints. 

Haskell, K.H. and Hanson, R.J., An algorithm for linear least-squares problems with equality and non-negativity constraints, Math. Prog., 21, 98-118, 1981. 

In PARAFAC modeling, non-negativity constraints were applied to all three dimensions. All analyses were performed with the N-way toolbox for MATLAB [30], which is a set of MATLAB routines designed to perform multi-way data analysis. 

An exploratory analysis performed by FSIW-EFA provides an estimate of the number of components in each pixel. For resolution purposes, only those pixels in the partial local rank map will be potentially constrained, because these are the pixels for which a robust estimation of the number of missing components can be obtained. However, the FSIW-EFA information is not sufficient to identify which components are absent from the constrained pixels. For identification purposes, the local rank information should be combined with reference spectral information, the ideal reference being the pure spectra of the constituents, although in most images not all of these are known. For the image components with no pure spectrum available, the reference taken is an approximation of this pure spectrum. These approximate pure spectra can be obtained by pure variable selection methods, or they may be the result of a simpler MCR-ALS analysis where only non-negativity constraints have been applied. 

This gives us a total of N partial derivative equations (19.66) and (19.67), one for each species in the system. We have c mass balance constraints (19.61), charge balance constraints (19.62), and N non-negative constraints (19.64). Solving these simultaneously with the N partial derivatives above gives us the necessary 2N+c+

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Thus, the non-negativity of solution is not an established constraint in the theoretical foundation of linear methods. On the other hand, the empirically formulated non-linear methods [Eqs. (55-56)] effectively secure positive and stable solutions. Such a weakness of the rigorous linear methods indicates a possible inadequacy in criteria employed for formulating the optimum solutions. In Section 6 we discuss possible revisions in assumptions employed for accounting for random noise in inversions. For example, it will be shown that by using log-normal noise assumptions the non-negativity constraints can be imposed into inversion in a fashion consistent with the presented approach inasmuch as one considers the solution as a noise optimization procedure. 

A simple and reliable way of constraining the input function is to apply the prescribed input function deconvolution method becanse this method allows the input function to be directly constrained. For example, a simple linear spline may be used as an input function. The non-negativity constraint is introdnced by a simple parameterization of the spline with parameters defined as the function valnes at the so-called knots where the linear line segments are joined. The inpnt fnnction will be non-negative by ensuring that all parameters, i.e., the function valnes at the knots are non-negative. 

A candidate AR construction method that utilizes hyperplanes to carve away unachievable space shall be discussed in Section 8.5.2. Linear constraints, such as non-negativity constraints on component concentrations and flow rates, may also be expressed in the form of a hyperplane equation. Hyperplanes therefore also arise in establishing bounds in state space. In Section 8.6, superstructure methods shall be described for the computation of candidate ARs. These methods, at their core, rely on the solution of a large... 

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