Originally posted by: SystemAdmin

[Gio said:]

Good evening to all

I have a convex optimization problem with few variables and constraints with a dual that become a simple QP that I can solve more efficiently that the primal. My question is Once the dual solution (y) is known can I evaluate the primal (x) ?

Tank to all
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#DecisionOptimization

Originally posted by: SystemAdmin

[gangooa said:]

One runs into the following dilemma while designing an approximation algorithm for an NP-hard optimization problem: for establishing the performance guarantee of the algorithm, the cost of the solution found needs to be compared with that of the optimal; however, computing the cost of the optimal is NP-hard as well. Hence a key consideration is establishing a good lower bound on the cost of the optimal solution, assuming we have a minimization problem. For optimization problems that can be expressed as integer programming problems, the following general methodology has been quite successful: use the cost of the optimal solution to the LP-relaxation as the lower bound. In fact, LP-duality not only provides a method of lower bounding the cost of the optimal solution, but also a general schema for designing the algorithm itself: the primal-dual schema. This schema has been applied to several problems including set cover and its generalizations [Jo, Lo, Ch, RV], the generalized Steiner network problem [AKR, GW, KR, WGMV, G+], and finding integral multicommodity flow and multicut in trees [GVY].
The primal-dual schema enables one to find special solutions to an LP, although so far it has only been used for obtaining good integral solutions to an LP-relaxation. In the past, the primal-dual schema has yielded the most efficient known algorithms to some cornerstone problems in P, including matching, network flow and shortest paths. These problems have the property that their LP-relaxations have optimal solutions that are integral, and so the primal-dual schema is able to find an optimal solution to the original integer program. Since numerous NP-hard optimization problems can be expressed as integer programs, this schema holds even more promise in the area of approximation algorithms.

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Originally posted by: SystemAdmin

[Julia2009 said:]

thanks for your information

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#DecisionOptimization

Originally posted by: SystemAdmin

[Julia2009 said:]

good to hearing



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Originally posted by: SystemAdmin

[chynna16 said:]

Thanks buddy Cool
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#DecisionOptimization

Originally posted by: SystemAdmin

[johncui said:]

“One runs into the following dilemma while designing an approximation algorithm for an NP-hard optimization problem: for establishing the performance guarantee of the algorithm, the cost of the solution found needs to be compared with that of the optimal; however, computing the cost of the optimal is NP-hard as well. Hence a key consideration is establishing a good lower bound on the cost of the optimal solution, assuming we have a minimization problem. For optimization problems that can be expressed as integer programming problems, the following general methodology has been quite successful: use the cost of the optimal solution to the LP-relaxation as the lower bound. In fact, LP-duality not only provides a method of lower bounding the cost of the optimal solution, but also a general schema for designing the algorithm itself: the primal-dual schema. This schema has been applied to several problems including set cover and its generalizations [Jo, Lo, Ch, RV], the generalized Steiner network problem [AKR, GW, KR, WGMV, G+], and finding integral multicommodity flow and multicut in trees [GVY].
The primal-dual schema enables one to find special solutions to an LP, although so far it has only been used for obtaining good integral solutions to an LP-relaxation. In the past, the primal-dual schema has yielded the most efficient known algorithms to some cornerstone problems in P, including matching, network flow and shortest paths. These problems have the property that their LP-relaxations have optimal solutions that are integral, and so the primal-dual schema is able to find an optimal solution to the original integer program. Since numerous NP-hard optimization problems can be expressed as integer programs, this schema holds even more promise in the area of approximation algorithms.”

The above message have solved the problem: "Once the dual solution (y) is known can I evaluate the primal (x) "???  in particular, in CPLEX?

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#DecisionOptimization

Originally posted by: SystemAdmin

[Abrahamlinkon said:]

The primal-dual schema enables one to find special solutions to an LP, although so far it has only been used for obtaining good integral solutions to an LP-relaxation. In the past, the primal-dual schema has yielded the most efficient known algorithms to some cornerstone problems in P, including matching, network flow and shortest paths. These problems have the property that their LP-relaxations have optimal solutions that are integral, and so the primal-dual schema is able to find an optimal solution to the original integer program. Since numerous NP-hard optimization problems can be expressed as integer programs, this schema holds even more promise in the area of approximation algorithms.

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