Abstract
During the past two decades, there have been active research for second-order cone programs (SOCPs) and second-order cone complementarity problems (SOCCPs). Various methods had been proposed which include the interior-point methods [1, 102, 109, 123, 146], the smoothing Newton methods [51, 63, 71], the semismooth Newton methods [86, 120], and the merit function methods [43, 48]. All of these methods are proposed by using some SOC complementarity function or merit function to reformulate the KKT optimality conditions as a nonsmooth (or smoothing) system of equations or an unconstrained minimization problem. In fact, such SOC complementarity functions or merit functions are closely connected to so-called SOC functions. In other words, studying SOC functions is crucial to dealing with SOCP and SOCCP, which is the main target of this chapter.
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Notes
- 1.
Sun and Sun did not consider the case of \(o(\Vert h\Vert )\) but their argument readily applies to this case.
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Chen, JS. (2019). SOC Functions . In: SOC Functions and Their Applications. Springer Optimization and Its Applications, vol 143. Springer, Singapore. https://doi.org/10.1007/978-981-13-4077-2_1
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DOI: https://doi.org/10.1007/978-981-13-4077-2_1
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