Penalty operators

Let K cV he a. convex closed subset of a reflexive Banach space V, I he a duality mapping, and P be a projection operator of V onto K. We are in a position to give a definition of a penalty operator. An operator (5 V V is called a penalty operator connected with the set K if the following conditions are fulfilled. Firstly, / is a monotonous bounded semicontinuous operator. Secondly, a kernel of / coincides with K, i.e. 

The proof of this theorem is based on the following lemmas. 

The right-hand side is nonnegative here, hence in view of the monotony of 0(t) the relation 

The first term of the right-hand side is nonnegative here in view of (1.101). The second term can be written in the form 

A boundedness and semicontinuity of the operator / follow from the same properties of I and P. The verification of the equivalence of conditions 

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As we can see, penalty operators can be built easier in Hilbert spaces. In applications H is often a Hilbert space such that... 

From the continuity of the penalty operator it follows that... 

Let us construct the standard penalty operator (5 v) = I v — Pv) and define the penalty problem depending on a small positive parameter s,... 

To linearize the penalty operator in (1.119) we use the following iteration scheme similar to (1.105),... 

Proof. We introduce the penalty operator p w) = — w — ) and consider the auxiliary boundary value problem with the positive parameter e > 0,... 

Proof. Let s, 5 be positive parameters, and p be the penalty operator introduced in the previous section. We consider the auxiliary problem... 

We now construct a penalized problem. To this end, define the penalty operator / —> by the formula... 

To construct a penalty problem, we introduce the penalty operator / —y H(Clc) by the relation... 

In so doing we have omitted the nonnegative term containing the penalty operator. Using the formula (5.181), the integration by parts can be done in the third and the fifth terms of the left-hand side of (5.189). Also, note that Mij satisfy equation (5.175). Integration of (5.189) in t from 0 to t results in the inequality... 

Kovtunenko V.A. (1997a) Iterative approximations of penalty operators. Numer. Funct. Anal, and Optimiz. 18 (3 4), 383-387. 

When a product fails to meet specification, it is either rejected as a waste or low-value product, burned as fuel, reprocessed, or blended with feed-stock or overpure product. The result is the use of more energy per unit of production. To avoid these penalties, operators typically make their products much purer than necessary, which also uses energy. Substantial savings can be achieved by being able to control product closer to specifications. This is the principal justification for advance control systems for evaporators. 

Sidney JB (1977) Optimal single-machine scheduling with earliness and tardiness penalties. Oper Res 25(l) 62-69... 

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