Like the matrix-valued functions used in solutions methods for semidefinite programs (SDPs) and semidefinite complementarity problems (SDCPs), the vector-valued functions associated with second-order cones are defined analogously and also used in solutions methods for second-order-cone programs (SOCPs) and second-order-cone complementarity problems (SOCCPs). In this article, we study further about these vector-valued functions associated with second-order cones (SOCs). In particular, we define the so-called SOC-convex and SOC-monotone functions for any given function . We discuss the SOC-convexity and SOC-monotonicity for some simple functions, e.g., <i>f</i>(<i>t</i>) = <i>t</i> ² <i>t</i> ³ 1/<i>t</i> <i>t</i> ^1/2, |<i>t</i>|, and [<i>t</i>]₊. Some characterizations of SOC-convex and SOC-monotone functions are studied, and some conjectures about the relationship between SOC-convex and SOC-monotone functions are proposed.

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It is well known that complementarity functions play an important role in dealing with complementarity problems. In this paper, we propose a few new classes of complementarity functions for nonlinear complementarity problems and second-order cone complementarity problems. The constructions of such new complementarity functions are based on discrete generalization which is a novel idea in contrast to the continuous generalization of FischerBurmeister function. Surprisingly, these new families of complementarity functions possess continuous differentiability even though they are discrete-oriented extensions. This feature enables that some methods like derivative-free algorithm can be employed directly for solving nonlinear complementarity problems and second-order cone complementarity problems. This is a new discovery to the literature and we believe that such new complementarity functions

Many optimization algorithms converge to stationary points. When the underlying problem is nonconvex, they may get trapped at local minimizers and occasionally stagnate near saddle points. We propose the Run-and-Inspect Method, which adds an “inspect” phase to existing algorithms that helps escape from non-global stationary points. The inspection samples a set of points in a radius R around the current point. When a sample point yields a sufficient decrease in the objective, we resume an existing algorithm from that point. If no sufficient decrease is found, the current point is called an approximate R-local minimizer. We show that an R-local minimizer is globally optimal, up to a specific error depending on R, if the objective function can be implicitly decomposed into a smooth convex function plus a restricted function that is possibly nonconvex, nonsmooth. Therefore, for such nonconvex objective functions, verifying global optimality is fundamentally easier. For high-dimensional problems, we introduce blockwise inspections to overcome the curse of dimensionality while still maintaining optimality bounds up to a factor equal to the number of blocks. Our method performs well on a set of artificial and realistic nonconvex problems by coupling with gradient descent, coordinate descent, EM, and prox-linear algorithms.