Convex functions linear case

The addition of inequality constraints complicates the optimization. These inequality constraints can form convex or nonconvex regions. If the region is nonconvex, then this means that the search can be attracted to a local optimum, even if the objective function is convex in the case of a minimization problem or concave in the case of a maximization problem. In the case that a set of inequality constraints is linear, the resulting region is always convex. 

The first one is that this particular form of H can also be used to prove the approach to equilibrium in the case of Boltzmann s kinetic equation for dilute gases. The Boltzmann equation is nonlinear and a different technique is needed to prove that all solutions tend to equilibrium. This technique is based on (5.6) other convex functions cannot be used. Incidentally, the Boltzmann equation is not a master equation for a probability density, but an evolution equation for the particle density in the six-dimensional one-particle phase space ( /i-space ). The linearized Boltzmann equation, however, has the same structure as a master equation (compare XIV.5). 

Remark 1 Note that the additional assumption makes the problem convex in y if y represent continuous variables. Ify Y = 0,1 , and they-variables participate linearly (i.e. f2,g2 are linear in y), then the relaxed master is convex. Therefore, this case represents an improvement over v3-GBD, and application of v3-GBD will result in valid support functions, which implies that the global optimum of (6.2) will be obtained. 

Depending on the form of the objective function, the final formulation obtained by replacing the nonlinear Eq. (17) by the set of linear inequalities corresponds to a MINLP (nonlinear objective), to a MIQP (quadratic objective) or to a MILP (linear objective). For the cases where the objective function is linear, solution to global optimal solution is guaranteed using currently available software. The same holds true for the more general case where the objective function is a convex function. 

The equality to zero is obtained only in the case where, for any i,j = 1,.. ., n we have dJNfii = 8jlN, i.e. when the vector 3 (with components 8t) is proportional to that with components N t, in other words there exists a value of X such that 6, = XNf)t. But this is possible only in the case in which all the components 8t are simultaneously either positive or negative. Since, at some non-zero value of x, the vectors with components N0t and Noi + 3t must lie in the same reaction polyhedron, the simultaneous positivity or negativity for all the Si values is forbidden by, for example, the law of conservation of the overall (taking into account its adsorption) gas mass = Zm,-(iV0i + x3t) 1.171 1 = 0, for any Af we have m, > 0, hence <5 cannot have the same signs. Consequently, in the reaction polyhedron, G is a strictly convex function since the sum of a strictly convex G0 with a linear function of Df and a strictly convex function of IV5 is strictly convex in this polyhedron. 

In the degenerate case we can have a situation, where a linear combination of ground-state densities is not necessarily itself a ground-state density. This has the consequence that the HK [equation (11)] and the Levy-Lieb [equation (38)] functionals are not necessarily convex, which for many applications is a disadvantage. A convex functional can be constructed by considering ensemble-v-representable (E-v-representable) densities [3,12,11]... 

Nonlinear optimization can be treated in special case if the objective function is convex (concave) and the feasible regimi is convex (cmicave). The most notable property is that a local minimum of a convex function on a convex feasible regimi is also a global minimum. Thanks to this, special optimization procedures can be developed. To note that linear functions are convex. 

It can be shown that this can be generalized to the case of more than two variables. The standard solution of a linear programming problem is then to define the comer points of the convex set and to select the one that yields the best value for the objective function. This is called the Simplex method. 

Convex Cases of NLP Problems Linear programs and quadratic programs are special cases of (3-85) that allow for more efficient solution, based on application of KKT conditions (3-88) through (3-91). Because these are convex problems, any locally optimal solution is a global solution. In particular, if the objective and constraint functions in (3-85) are linear, then the following linear program (LP)... 

L(x,y, Xk,(ik), L(x ,y,X ,fil) represent local linearizations around the points xk andxk of the support functions (y Ak,pk), (y A1, p.1), respectively. Therefore, the aforementioned assumption is valid if the projected problem v(y) is convex in y. If, however, the projected problem u(y) is nonconvex, then the assumption does not hold, and the algorithm may terminate at a local (nonglobal) solution or even at a nonstationary point. This analysis was first presented by Floudas and Visweswaran (1990) and later by Sahinidis and Grossmann (1991), and Bagajewicz and Manousiouthakis (1991). Figure 6.1 shows the case in which the assumption is valid, while Figure 6.2 shows a case for which a local solution or a nonstationary point may result. 

This paper presents a general mathematical programming formulation the can be used to obtain customized tuning for PID controllers. A reformulation of the initial NLP problem is presented that transforms the nonlinear formulation to a linear one. In the cases where the objective function is convex then the resulting formulation can be solved easily to global optimality. The usefulness of the proposed formulation is demonstrated in five case studies where some of the most commonly used models in the process industry are employed. It was shown that the proposed methodology offers closed loop performance that is comparable to the one... 

For the n-dimensional case, the region that is defined by the set of hyperplanes resulting from the linear constraints represents a convex set of all points which satisfy the constraints of the problem. If this is a bounded set, the enclosed space is a convex polyhedron, and, for the case of monotonically increasing or decreasing values of the objective function, the maximum or minimum value of the objective function will always be associated with a vertex... 

Figure 2.6 summarizes the relationship between the isotherm function in the case of a pure component, q = /(C) in Eq. 2.4, and the band profile. The top row (Figures 2.6a,f,k) shows the three major t)q)es of isotherms encoimtered in chromatographic systems linear (a), convex upward or Langmuirian (f), and convex... 

In the second step of the utihty model, the input values are evaluated. For this purpose the decision matrix is transformed into a consequence matrix. So the input values are transformed into the part-worth utilities. Thus the consequence matrix consists only of the part-worth utility values. In order to perform this transformation, the utility model assumes that the decision maker has a precise idea about the utility placing to criterion s characteristics. Based on his preferences, different transformation functions can be considered, linear, concave and convex. In case the decision maker has a clear preference in respect of utility allocation, as in the present case study, the use of the utility model is possible. However specifying the preferences for all criteria and related characteristics in industrial business is not always easy (Aven 2007). On this note when clear preferences cannot be worked out, application of the utility model is not reasonable. 

Since the conditions in general are difficult to verify, Thowsen discusses special cases under which the (y,p) policy is optimal for the problem considered. For instance, under a linear expected demand curve, linear stockout costs, convex holding costs, and a demand distribution that is a PF2 distribution, the iViP) policy is optimal. Furthermore, if excess demand is backlogged, the demand curve is concave and the revenue is collected a fixed number of periods after the time orders are placed, then no assumptions are needed on the cost and demand distribution for optimality of the critical number policy, and for this case the decision on price and quantity decisions can be made separately. Thowsen also shows that if negative demand is disallowed, the optimal price will be a decreasing function of increasing initial inventory. 

The performed analysis of problems solved by using MEIS has shown the possibilities for their reduction to convex programming (CP) problems in many important cases. Such reduction is often associated with approximation of dependences among variables. There are cases of multivalued solutions to the formulated CP problems, when the linear objective function is parallel to one of the linear part of set D y). Naturally the problems with non-convex objective functions or non-convex attainability sets became irreducible to CP. Non-convexity of the latter can occur at setting kinetic constraints by a system of linear inequalities, p>art of which is specified not for the whole region D (y), but its individual zones. 

Also, note that for the case of linear systems considered in this work, only multihnear terms appear as the non-convex terms in the problem formulation. Detailed description of the convex underestimators for trilinear and multilinear functions in general can be found in [20]. 

The above consideration may be extended to systems in which R linearly independent reactions are taking place. In this case the free enthalpy of the system is a function of R variables iu 2 Convexity of the function G = G(( i, 2 < r)... 

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