Convex Optimization Without Slater Condition

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SUMMARY

In convex optimization, the Slater condition ensures that the dual optimum equals the primal optimum. When the Slater condition does not hold, a dual gap exists, complicating the optimization process. To address this, practitioners often apply convexification techniques to the constraints of nonlinear nonconvex optimization problems. Despite the presence of a dual gap, methods exist to determine the primal optimum value.

PREREQUISITES
  • Understanding of convex optimization principles
  • Familiarity with the Slater condition
  • Knowledge of duality in optimization
  • Experience with convexification techniques
NEXT STEPS
  • Research methods for convexifying nonlinear nonconvex optimization constraints
  • Study duality theory in depth, focusing on cases without the Slater condition
  • Explore practical applications of convex optimization in real-world scenarios
  • Review academic literature on primal-dual algorithms for optimization
USEFUL FOR

Mathematicians, optimization researchers, and engineers working on complex optimization problems, particularly those dealing with nonlinear nonconvex scenarios.

mertcan
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Hi, initially I am aware of the fact that when slater condition holds, then dual optimum equals primal optimum in convex optimization. But if slater condition does not hold then dual gap exist. When we have nonlinear nonconvex optimization we apply convexification of constraints including different methods. Actually we have to use convex optimization whereas we have nonlinear nonconvex optimization. So, Even we have some dual gap in our convex optimization how we can find the primal optimum value?
 
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