Logarithmic barrier decomposition-based interior point methods for stochastic symmetric programming

Decomposition-based interior point methods for stochastic quadratic second-order cone programming

“…The proof follows from Lemma 2.1, and the fact that Q p ðÊ þ Þ ¼Ê þ , and likewise, as an operator, Q p ðintðÊ þ ÞÞ ¼ intðÊ þ Þ, becauseÊ þ is a symmetric cone [15]. h Now, with this change of variables, Problem (6) and (7) becomes min e gðl; xÞ :¼ 1 Observe that Problems (7) and (12) have the same minimizer but their optimal objective values are equal up to the constant 2l ln det p (as detðQ p qÞ ¼ det 2 ðpÞ detðqÞ for any two elements p; q 2 E m [16, Chapters II]).…”

“…Because y ðkÞ and s ðkÞ may not operator commute (see Appendix A for definition), the statement that y ðkÞ s ðkÞ ¼ l e 2 implies y ðkÞ ¼ l s ðkÞ À1 may not true [15,Remark 1]. Also, for the same reason, the equality detðy ðkÞ s ðkÞ Þ ¼ det y ðkÞ det s ðkÞ may not hold.…”

“…It is easy to see that the quadratic function 1 2 x T Gx þ c T x is a l strongly self-concordant barrier on F 1 . In view of [15,Lemma 4], the function Àl ln det x is also a l strongly self-concordant barrier on F 1 . By Lemma …”

“…For this purpose, we first give upper bounds on /ðl; xÞ and / 0 ðl; xÞ respectively. We have the following lemma which is due to Zhao [9, Lemma 7] (see also [15,Lemma 10]). We now prove the following lemma.…”

“…Our procedure in this paper closely follows that of [9,15], but our setting is clearly different and promising to be applicable to a different and wide variety of real-world applications. The use of convex quadratic objective together with secondorder cone constraints provides an important more general problem setting than the one studied in [12] and other papers.…”