Let us consider the following optimization problem $$ \min \,\,\, (1/2)x^{T}Px + q^{T}x $$ $$ \text{subject to} \quad Ax \in C $$ where $C$ is a closed convex set.
Assume now that $p$ is positive semidefinite matrix. Therefore, the solution might not be unique.
Next, according to CRAN Task View: Optimization and Mathematical Programming, the problem can be approached by several optimisation methods, i.e. there are methods which let $P$ be positive semidefinite.
I do not understand, is there guarantee that a solution (one of, if there are many) found by the methods will provide the minimum of the objective function?