Penalty term

In some applications it may be necessary to prescribe a pressure datum at a node at the domain boundary. Although pressure has been eliminated from the working equations in the penalty scheme it can be reintroduced through the penalty terms appearing in the boundary line integrals. 

In this case the boundary conditions (5.81) are included in (5.84). At the first step we get a priori estimates. Assume that the solutions of (5.79)-(5.82) are smooth enough. Multiply (5.79), (5.80) by Vi, Oij — ij, respectively, and integrate over fl. Taking into account that the penalty term is nonnegative this provides the inequality... 

Integrate by parts in the fifth and sixth terms of the left-hand side of (5.152) taking into account the boundary conditions (5.149)-(5.151) and the Green formula like (5.138) for the domain fic- The penalty term is nonnegative and satisfy the equation (5.144). Hence the uniform in the s,5 estimate follows. 

A derivation of the next estimate requires the boundedness of the penalty term in L Q), i.e. uniformly in e,6... 

Now we can differentiate the equations (5.334)-(5.337) with respect to t and multiply by Vt, Wt, nt, rrit, respectively. The penalty terms are nonnegative and, therefore, they can be neglected. As a result, the following differential inequality is derived ... 

In the opposite limit, NA NB, the corona is thin and similar in structure to a flat grafted layer. The dominant penalty term in such crew-cut micelles is Fcore. Accordingly,... 

The second term in Eq. (1) is a penalty term representing constraints on the control field e(r) via a functional / and is extremely important in determining the outcome of the optimization. The most common penalty term, and that first introduced by Rabitz and co-workers [41], is... 

In Appendix A, we follow the derivation of Shi and Rabitz and carry out the functional variation of the objective functional [Eq. (1)] so as to obtain the equations that must be obeyed by the wave function (vl/(t)), the undetermined Lagrange multiplier (x(0)> the electric field (e(t)). Since the results discussed in Section IV.B focus on controlled excitation of H2, where molecular polarizability must be considered, the penalty term given by Eq. (3) is used and the equations that must be obeyed by these functions are (see Appendix A for a detailed derivation) ... 

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] ... 

An alternate method for introducing pulse restrictions has been introduced by one of us (AB) recently within an iterative scheme for solving the optimal control equations [70], The idea is that a new reference field e(t) is constructed based on the field from the previous iteration after the application of a filter function F to ensure the fulfilment of some predesigned temporal and spectral properties. Therefore, a penalty term of the form... 

With the objective functional, J, being defined as in Eq. (1) and using the penalty term/[e(f)] = the gradient of the objective functional with respect to... 

If we had used the penalty term p we would have obtained a similar... 

The parameter e(t) has now canceled out on the two sides of Eq. (A.13) and the equation is no longer an equation for e(t). Our conclusion is that it is not appropriate to use the standard form for the fluence penalty term if the laser molecule interaction contains a significant contribution from the molecular polarizabUty. 

A matrix will be used to determine which resource can be used for the production of a product. The cost statement is some kind of penalty term. (Example the production of a product on a certain resource is more expensive than on another resource due to higher energy consumption.)... 

The main idea of the algorithm of Caroe and Schultz [11] is to decompose a 2S-MILP into its scenarios by Lagrangian relaxation of the non-anticipativity constraints. In a Lagrangian relaxation, constraints are removed and included in the objective function with a penalty term. 

The explicit feasibility constraints of (MASTER) are given by the linear first-stage constraints in (9.4.2). In a classical penalty function approach, the explicit feasibility constraints are relaxed while the violation of these constraints is considered by an additional penalty term in the fitness function. However, this method would waste valuable CPU time since the MILP subproblems (SUB) have to be solved also for the fitness evaluation of infeasible individuals. A similar method which does not require the solution of the MILP subproblems for infeasible individuals is the use of a modified objective Junction that separates the objective and the feasibility... 

The parameter/max denotes a conservative upper bound off (x). For the 2S-MILP it is easily calculated by maximizing the integer relaxation of (DEP). A positive penalty term p(x) is used to measure the amount of infeasibility. This steers the search in infeasible regions towards the feasible region. The penalty for the violation of the first-stage constraints is provided by ... 

Barrier methods are not directly applicable to problems with equality constraints, but equality constraints can be incorporated using a penalty term and inequalities can use a barrier term, leading to a mixed penalty-barrier method. 

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