Feasibility problems

Is a Proposed Microbiological Transformation Thermodynamically Feasible Problem... 

To include all mentioned possibilities Fletcher and Leyffer (1994) formulated a general feasibility problem (FP) defined as... 

The solution of the feasibility problem (FP) provides information on the Lagrange multipliers for the equality and inequality constraints which are denoted as Ak,fik respectively. Then, the Lagrange function resulting from on infeasible primal problem at iteration k can be defined as... 

The primal does not have a feasible solution for y = y. Solve a feasibility problem (e.g., the / minimization) to determine the multiplier vectors A,(i of the feasibility problem. 

Similarly, we assume that we can use the optimal solution of the feasibility problem (if the primal is infeasible) for the determination of the support function (y A k,pk). 

Remark 1 In the case of infeasible primal problem, we need to solve a feasibility problem. One way of formulating this feasibility problem is the following ... 

Remark 3 Kocis and Grossmann (1989a) suggested another alternative formulation of the feasibility problem, in which a penalty-type contribution is added to the objective function that is,... 

A feasible solution to the primal problem exists when the penalty term is driven to zero. If the primal does not have a feasible solution, then the solution of problem (6.31) corresponds to minimizing the maximum violation of the inequality constraints (nonlinear and linear in jc). A general analysis of the different types of feasibility problem is presented is section 6.3.3.1. 

Alternative (ii) Add to the relaxed master problem the linearizations around the infeasible continuous point. Note though that to treat the relaxed master problem we need to have information on the Lagrange multipliers. To obtain such information, a feasibility problem needs to be solved and Viswanathan and Grossmann (1990) suggested one formulation of the feasibility problem that is,... 

If the primal problem P(yk) is infeasible, then the following general feasibility problem was proposed by Fletcher and Leyffer (1994) ... 

Remark 1 Note that the general feasibility problem (6.41) has in the objective function a summation over only the infeasible inequality constraints. Note also that a similar approach was presented in section 10.3.3.1. Fletcher and Leyffer (1994) proved the following important property for problem (6.41). 

Remark 7 The constraints (6.48) define the set of y G Y D V, and hence we can now formulate the master problem correctly. Note that we replace the sety G Y n V with constraints (6.48) that are the outer approximations at the points yk for which the primal is infeasible and the feasibility problem (6.41) has as solution xk. 

Remark 1 The first condition of CTP-e and the CTD-e are classified as e-value convergence tests since they correspond to feasible problems. The second condition of CTP-e and the CTDU-e are denoted as e-feasibility convergence tests since they correspond to feasibility problems. 

The extensive options allow the user to tune the performance of each algorithm. Some of the frequently used options are (i) the incorporation of integer cuts, (ii) the solution of continuous relaxation problems, (iii) alternative feasibility problems, (iv) an automatic initialization procedure, (v) a tightening of bounds procedure, and (vi) solvers parameter changes. 

The simplest kind of feasibility problem arises when the properties constrained are linear functions of the operating variables, of the following form ... 

Xi, x2,. . . , Xa being the operating variables, and oi, aa,. . . an, and b known constants. Fortunately, such linear inequalities describe most industrial constraints fairly well. In the chemical process industries, constraints usually involve stoichiometric relations, which are, of course, linear, or such physical properties as vapor pressure which can often be approximated by linear functions like Raoult s Law. Most of the present methods for solving feasibility problems only work when the constraints are linear. Simple as it is, the linear model has many applications,... 

When the criterion optimized is a linear function of the operating variables, the feasibility problem is said to be one in linear programming. Being the simplest possible feasibility problem, it was the first one studied, and the publication in 1951 of G. Dantzig s simplex method for solving linear-programming problems (D2) marked the beginning of contemporary research in optimization theory. Section IV,C is devoted to this important technique. 

Stopped when the lower and upper bounds are within a specified tolerance. Also, if infeasible NLP subproblems are found, the feasibility problem (NLPF) is solved to provide the point x (commonly NLPF—oo). The OA method generally requires relatively few cycles or major iterations. It trivially converges in one iteration ff x,y) and gix,y) are linear. It is also important to note that the MILP master problem need not be solved to optimality. In fact, given the upper bound UB and a tolerance e, it is sufficient to generate the new (y by solving... 

The Maxwellian molecules are useful in exploratory calculations in which a differentiable potential function is needed. For these molecules [66] the in-termolecular force between pairs at a distance r apart is of the form where K is a constant. Adopting this particular potential representation the solution of the equation of transfer reduces to a feasible problem, thereby Maxwell [66] obtained analytical expressions for the transport coefficients as mentioned earlier. 

Capacity planning verifies the feasibility of the plans at both aggregate and detailed levels. At first stage, master production schedule feasibility can be verified in an aggregated way. After material planning is performed, capacity planning verification may induce adjustments in the detailed plans. Also, the master production schedule can be modified if capacity planning reveals feasibility problems. 

Although the SOS relaxations apply to the original problem in (24), they are more easily described for a modified feasibility problem given nxn symmetric matrices Mq, Mi,. .., determine if the relation... 

Check if there are any constraint violations for all 6 G [0.45,0.5]. Note that there are no operating variables in this process since the output set-points are treated as design variables. Hence, the feasibility problem (3) reduces to ... 

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