Implicit constraints

As first-stage feasible solutions in general do not necessarily have a feasible completion in the second-stage due to the implicit constraints in (SUB), the total set of feasible solutions for x is a subset of the first-stage feasible solutions. In this case, the program is called a 2S-MILP without relative complete recourse. For a 2S-MILP with relative complete recourse, each first stage feasible solution x has a feasible completion in the second-stage. 

Inequality constraints. Implicit constraints exist because of the use of dimensionless variables... 

The constraints changed from one trial configuration of the reaction system to the next, but typically included things like the minimum coolant temperature to permit efficient utilization of the heat of reaction as process steam, the maximum allowable aldehyde concentration in the condensed crude product to avoid refining and product specification problems, and a prescribed reactor pressure drop to insure adequate flow distribution among the reactor tubes at a minimum energy cost. All of these are implicit constraints — they establish the maximum or minimum levels for certain response variables. Explicit constraints comprise the ranges for search variables. 

Retain in memory and recall as needed the implicit constraints in a problem. 

Remember that implicit constraints (equations) exist in the problem formulation because of the definition of mass fraction namely, that the sum of the mass fractions in each stream must be unity ... 

What is the major category of implicit constraints (equations) you encounter in material balance problems ... 

The equations of motion of a structure with dampers are treated as additional implicit constraints. Moreover, itis assumed that the damper s damping factors are continuous design variables. However, in practical applications, the damper s capacity and size can be found only from a set of actually manufactured dampers. Dampers are fixed to a structure with the help of braces, which are treated as elastic elements or as rigid elements when the shear fiame is used as the model of a real structure. 

For agiven set of m possible damper locations, find the positions of dampers and the value of their main damping factors. which minimize the objective function (1) and fulfill the explicit constraints of Equation (2) and other implicit constraints mentioned above. 

Let s now relax the requirement that predicate r be binary. But we keep the (so far implicit) constraint that the induction parameter be simple. Supposing predicate r is -ary (where n is a schema-variable), this new setting implies that Y becomes a vector Y ofn-1 variables Yj. and that vector TY becomes a vector TY of n-l vectors TKy, each of which is a vector of t variables TYji (where j, I are notation-variables). Similarly, HY becomes a vector HY of n-l vectors HYj, each of which is a vector of h j) variables HYji, where h /l is a schema function-variable. Thus ... 

You should be aware of one important difference between the electrochemical potential and the chemical potential. The chemical potential describes the free energy of inserting one particle into a particular place or phase, subject to any appropriate constraint. Constraints are introduced explicitly. In contrast, the electrochemical potential always carries an implicit constraint with it overall electroneutrality must be obeyed. This is a very strong constraint. You can never insert a single ion in a volume of macroscopic dimensions because that would violate electroneutrality. You can insert only an electroneutral combination of ions. 

Although the constraints on the amino nitrogen in the formulation above can be considered directly as problem variables, the C coordinates are not explicit variables and consequently must be defined as a function of the other variables [234]. Because the energy minimization problem described above involves these implicit constraints on the location of C, a penalty function must be added to the function E in order to implement these constraints. The modified form of the function E is then [234] ... 

In the top-down design phase, we identified an additional, implicit constraint to be addressed that is not part of the design specifications. Specifically, the flash unit processes a pseudobinary mixture (A/C and B/D), and thus, its temperature and pressure cannot be specified independently. This design issue has been handled by controlling the flash unit feed temperature (flash unit preheater exit temperature) Tpp mther than itself. 

An implicit edge process is involved in the regularization process where A acts as a scale parameter which gives a constraint on the size of the homogeneous patches and p. comes from ho = -y/ p/A where ho is the threshold above which a discontinuity is introduced. We propose, then to combine these two functionals to obtain a satisfactory solution ... 

The form of the Hamiltonian impedes efficient symplectic discretization. While symplectic discretization of the general constrained Hamiltonian system is possible using, e.g., the methods of Jay [19], these methods will require the solution of a nontrivial nonlinear system of equations at each step which can be quite costly. An alternative approach is described in [10] ( impetus-striction ) which essentially converts the Lagrange multiplier for the constraint to a differential equation before solving the entire system with implicit midpoint this method also appears to be quite costly on a per-step basis. 

Develop via mathematical expressions a valid process or equipment model that relates the input-output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 

In early work in the optimal control theory design of laser helds to achieve desired transformations, the optimal control equations were solved directly, without constraints other than those imposed implicitly by the inclusion of a penalty term on the laser huence [see Eq. (1)]. This inevitably led to laser helds that suddenly increased from very small to large values near the start of the laser pulse. However, physically realistic laser helds should tum-on and -off smoothly. Therefore, during the optimization the held is not allowed to vary freely but is rather expressed in the form [60] ... 

For practical purposes, implicit schemes are the methods of choice when the solution is smooth and well behaved as a function of time. In that case much larger time steps can be taken than with explicit schemes, thus allowing a reduction in the computational effort. When large temporal gradients and rapid variations are expected, accuracy constraints set severe limits to the time-step size. In that case explicit schemes might be favorable, as they come with a reduced numerical effort per time step. 

According to Cano-Ruiz and McRae, process design as a part of process development starts with problem framing. At this stage, the concept definition, scope of analysis, design objectives, constraints, evaluation and interruption criteria have to be determined. They argued that framing decisions are often made implicitly, whereas the critical importance... 

It appears thus, that writing E as C+C where CC+ = ln is already taking account of the first constraint of hermiticityof P, and also includes the two conditions P2 = P and Rank P=N. That is to say, the hermiticity constraint is implicitly taken into account as soon as one decomposes P as C+C, the two other constraints being completely summarized by CC+ = In- It seems therefore not necessary to superimpose the hermiticity constraint on CC+ = In, for it has already been done. 

Secondly, the network layout showed in Fig. 12.7 shows that 12.5 t of water should be supplied to Process 3, instead of 25 t stipulated in the problem specification. This can only be true if this process does not have flowrate constraints, but has a fixed mass load. The assumption of fixed mass loads was never mentioned in the analysis. This variation of flowrate is contrary to the assumption made in targeting. During targeting it was implicitly assumed that the flowrates were fixed as shown by the calculation of water demand in each of the time subintervals. 

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