Convex Optimization Without Slater Condition

In summary, the Slater condition is a constraint qualification that guarantees the existence of a strictly feasible point in convex optimization problems. It is important because it ensures the existence of a global optimal solution. While convex optimization problems can be solved without satisfying the Slater condition, the optimal solution may not be unique and may depend on the starting point of the optimization algorithm. The Slater condition can be checked by verifying if the constraints are strictly feasible, and techniques such as interior-point methods and subgradient methods can be used to solve convex optimization problems without satisfying the condition. However, these techniques may not guarantee global optimality.
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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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Related to Convex Optimization Without Slater Condition

What is the Slater condition in convex optimization?

The Slater condition, also known as the strict feasibility condition, is a necessary condition for a convex optimization problem to have a finite optimal solution. It states that there exists a strictly feasible point, which is a point that satisfies all the constraints strictly, not just at the boundary.

What is the significance of the Slater condition in convex optimization?

The Slater condition is important because it guarantees the existence of a finite optimal solution for a convex optimization problem. This means that the problem can be solved efficiently using various optimization algorithms.

What happens if the Slater condition is not satisfied in a convex optimization problem?

If the Slater condition is not satisfied, it means that there is no strictly feasible point and the problem may not have a finite optimal solution. In this case, the problem may need to be reformulated or solved using specialized algorithms for problems without a strictly feasible point.

Can the Slater condition be relaxed in convex optimization?

Yes, the Slater condition can be relaxed by introducing a tolerance parameter. This means that the problem does not need to have a strictly feasible point, but instead, a point that satisfies the constraints within a certain tolerance level. This allows for a wider range of problems to be solved using convex optimization techniques.

What are some applications of convex optimization without the Slater condition?

Convex optimization without the Slater condition has various applications in fields such as machine learning, signal processing, and finance. It is commonly used in problems with non-strict constraints, such as portfolio optimization and support vector machines. It is also used in problems with noisy or uncertain data, where the Slater condition may not hold.

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