Bildhauer M.'s Convex variational problems PDF

By Bildhauer M.

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Example text

It is therefore impossible to solve the problem optimally using this modeling idea. But, the model is helpful for computing an upper bound for the optimum objective function value of an instance. e. 7) for 1 E {1, ... , L} and s E Sl. 7). The optimum objective function value of the LP-relaxation defines an upper bound. Although the LP-relaxation of the model formulation still has a huge number of columns, it can be solved optimally using column generation techniques. 4. 7). The trick is that not all columns need to be considered.

In more detail, phase one can be specified as follows. It maintains a set U of activities which have not been scheduled so far. It begins at time t = 0 and determines the set D ~ U of activities which may be started at this time without violating the precedence constraints, the resource constraints, and the deadline. If more than one activity is contained in D, 7r is used to select the activity to be scheduled. At time t, activities 48 The Net Present Value are scheduled one after the other until no further activities can be started at t.

By definition, the activities in a chain are in a total order given by the set E of precedence constraints. This gives the motivation to investigate the following idea. Roughly speaking, we will decompose the network for a given project into 1 = 1, ... , L chains. Each of these chains will then be scheduled separately. If these schedules fit together, we have a solution for the overall problem. 2. Creating Chains Let us start with addressiI;tg issues related to decomposing a network into chains.

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Convex variational problems by Bildhauer M.


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