Bolor Jargalsaikhan (Universität Groningen, NL) - Genericity and stability in linear conic programming

In linear conic programming, we maximize or minimize a linear function over the intersection of an affine space and a convex cone.  A property is said to be stable at a problem instance if the property still holds under a small perturbation of the problem data. We say that a property is weakly generic if the property holds for almost all problem instances.

We start by showing that Slater's condition is weakly generic and stable. It is known that uniqueness of the optimal solution, nondegeneracy, strict complementarity are weakly generic properties in conic programming.

We then investigate the stability of these weakly generic properties. For the semidefinite programs, we show that all these weakly generic properties are stable. Moreover, we characterize first order optimal solutions in conic programs and give necessary and sufficient conditions for its stability.