Second-smallest-value conjecture for CDT subproblems and quartically constrained quadratic optimization

A CDT subproblem is a cubic-regularized optimization subproblem, and a nonconvex quadratic optimization problem with a quartic constraint is an optimization problem whose objective is nonconvex quadratic and whose constraint has degree four. A local nonglobal minimum is a local minimizer that is not globally minimizing, and the smallest local nonglobal minimum is the one with the least objective value among all such minima. Second-smallest-value conjecture. For a CDT subproblem or a nonconvex quadratic optimization problem with a quartic constraint, the smallest local nonglobal minimum has the second smallest objective function value among all KKT points. This conjecture proposes extending the paper's second-smallest-value result from trust-region and $p$-regularized subproblems to these more general optimization problems; the source does not establish whether the extension holds.