By Stephan Dempe
Bilevel programming difficulties are hierarchical optimization difficulties the place the restrictions of 1 challenge (the so-called higher point challenge) are outlined partly by way of a moment parametric optimization challenge (the decrease point problem). If the reduce point challenge has a precise optimum answer for all parameter values, this challenge is such as a one-level optimization challenge having an implicitly outlined goal functionality. specified emphasize within the ebook is on difficulties having non-unique reduce point optimum options, the confident (or vulnerable) and the pessimistic (or robust) methods are mentioned. The publication starts off with the mandatory leads to parametric nonlinear optimization. this is often through the most theoretical effects together with valuable and enough optimality stipulations and answer algorithms for bilevel difficulties. Stationarity stipulations can be utilized to the reduce point challenge to rework the positive bilevel programming challenge right into a one-level challenge. homes of the ensuing challenge are highlighted and its relation to the bilevel challenge is investigated. balance homes, numerical complexity, and difficulties having extra integrality stipulations at the variables also are mentioned. viewers: utilized mathematicians and economists operating in optimization, operations study, and financial modelling. scholars attracted to optimization also will locate this ebook helpful.
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Extra info for Foundations of Bilevel Programming (Nonconvex Optimization and Its Applications)
It is the most important question in this circumstance if the classes and coincide or not with the negative answer assumed to be true by almost all scientists. The optimization problems corresponding to the decision problems in the class are members of the class of -hard problems. The 0/1 knapsack and the traveling salesman problems are -hard, their decision variants are -complete. A large number of Linear bilevel problems 49 -complete and -hard problems can be found in the monograph . For showing that some problem II is -complete another problem has to be found which is known to be -complete and to prove that this problem is a “subproblem” of II.
The last approach is a condition for a global optimal solution of the bilevel programming problem and uses linear programming duality to formulate a parametric nonconvex quadratic optimization problem. It is shown that the optimal function value of this problem is equal to zero if and only if an optimal solution corresponds to a feasible solution for the bilevel problem. Thus, global optimality for the bilevel problem is related to the lower bound for the parameter values leading to zero optimal function values of the quadratic problem.
In this case, the second step is not too expensive. 8. On the other hand, if a descent direction can be found than stepping forward along this direction a new parameter value is obtained for which a new basic matrix of appears. An algorithm of this type has first been developed in . 9) and that the restriction to equations in the lower level problem is no loss of generality. Details are left to the interested reader. 7). 7) : with sufficiently large. 27) has a solution. 13 () Let the set be nonempty and bounded.