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This paper deals with a minimax control problem for semilinear elliptic variational inequalities associated with bilateral constraints. The control domain is not necessarily convex. The cost functional, which is to be minimised, is the sup norm of some function of the state and the control. The major novelty of such a problem lies in the simultaneous presence of the nonsmooth state equation variational inequality and the nonsmooth cost functional the sup norm.
In this paper, the existence conditions and the Pontryagin-type necessary conditions for optimal controls are established. To send this article to your Kindle, first ensure no-reply cambridge. Find out more about sending to your Kindle.
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This article has been cited by the following publications. This list is generated based on data provided by CrossRef. Chen, Qihong and Ye, Yuquan Numerical Functional Analysis and Optimization, Vol. Chen, Qihong Optimal obstacle control problem for semilinear evolutionary bilateral variational inequalities. In some special cases, bilevel programs can be modeled as integer programs, though, but I guess that most of the meaningful bilevel programs cannot. Your approach with using the auxiliary C variable should work, so it seems that currently you are facing an OPL modeling problem.
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But you should be aware of the fact that your approach requires one constraint for every single value in the domain of Z. So, the approach will only be practical if Z is small, in particular finite. Therefore, z cannot be continuous, must be bounded, and if z is a vector instead of a single variable, then it should better be of small dimension to avoid the combinatorial explosion in the domain. Dear Daniel and Tobias, thank you for the answers! Daniel, my problem is a little bit different. I describe it below.
Tobias, do you have any hints - how this bilevel models could be modeled and solved? I have tried to implement a solution for a little bit more complicated problem tiny even though and find out that it does not work in my case works incorrectly. Imagine, there is a warehouse and I have to decide about orders of different products at the beginning of each period.
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I have ordering and holding costs. However, demand is not known exactly - only upper and lower bounds are given. My goal is to optimize the worst case - to solve the minimax problem. I attached the SmallModel file wich contains the initial model and modified one below here I have tried to implement the proposed apparoach with additional variable C.
Problem is that I want to check all possible combinations of demands in each period, when each of them changes in fixed borders borders could be different. The proposed approach works incorrectly however: it just finds the maximal possible value of demand through all periods and takes it as a value in all planning periods.
Another guess is that even if I implement the approach, I can't simply check all possible combinations of demands in each planning period, because this is a tremendous amount depending on problem size as it has been already said by Tobias. So I need some optimizational algorithm to find the inner maximum. But I can't believe that this tiny study example could not be solved by the ILOG - the proffessional optimization software. I guess that I just do not know how to do this.
CPLEX can only deal with convex sets plus integrality, but minimax problems do not necessarily have this property. What should be possible is to use a cutting plane approach Bender's decomposition , where you introduce a new auxiliary variable for the objective of the inner maximization problem. Then, when CPLEX presents you a solution for the outer variables, you solve the inner maximization problem and find the value for the auxiliary objective variable.
But I am not sure whether this would really give a useful solver for larger problems.