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Computation Algorithm for Convex Semi-infinite Program with Second-Order Cones: Special Analyses for Affine and Quadratic Case

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Abstract

We focus on the convex semi-infinite program with second-order cone constraints (for short, SOCCSIP), which has wide applications such as filter design, robust optimization, and so on. For solving the SOCCSIP, we propose an explicit exchange method, and prove that the algorithm terminates in a finite number of iterations. In the convergence analysis, we do not need to use the special structure of second-order cone (SOC) when the objective or constraint function is strictly convex. However, if both of them are non-strictly convex and constraint function is affine or quadratic, then we have to utilize the SOC complementarity conditions and the spectral factorization techniques associated with Euclidean Jordan algebra. We also show that the obtained output is an approximate optimum of SOCCSIP. We report some numerical results involving the application to the robust optimization in the classical convex semi-infinite program.

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Notes

  1. Function f and cone \(\mathcal{K}\) are convex. Moreover, the KKT condisions of the problem “min f(x) s.t. \(x\in \mathcal{K}\)” can be written as \(x\in \mathcal{K}\), \(\nabla f(x)\in \mathcal{K}\) and \(x^{\top }\nabla f(x)=0\).

  2. Assumption A (iii) and Proposition 4.1 yield \(0<-c({\overline{x}},\overline{s}_{\max })\le c({x^k},s^k)-c({\overline{x}},s^k)\le \nabla _{x}c({x^k},s^k)^\top ({x^k}-{\overline{x}})\le \Vert \nabla _{x}c({x^k},s^k)\Vert (M+\Vert {\overline{x}}\Vert )\), where \(\overline{s}_{\max }:={\mathrm{argmax}}_{s\in \Omega }c({\overline{x}},s)\). Therefore, we can choose \(\gamma :=-c({\overline{x}},\overline{s}_{\max })/(M+\Vert {\overline{x}}\Vert )>0\).

  3. When \(\mathcal{I}^{k}_2=\emptyset \), we immediately obtain the desired result \(\mathcal{I}^{k}_2\subseteq \mathcal{I}^{k+1}_2\).

  4. When \(\Omega =[l_s,u_s]\subset \mathbb {R}\), we test the 101 points \(l_s+i(u_s-l_s)/100\) with \(i=0,1,\ldots ,100\). When \(\Omega =[l_s,u_s]^2\subset \mathbb {R}^2\), we test the 121 points \((l_s+i(u_s-l_s)/10,\,l_s+j(u_s-l_s)/10)^\top \) with \(i=0,1,\ldots ,10\) and \(j=0,1,\ldots ,10\).

  5. For Problem 7, the final active index is located in the interior of \(\Omega \), and for other six problems (5, 6, 8, 10, 11, 12), they are located on the non-vertex boundary.

  6. For example, when \(\mathcal{K}=(\mathcal{K}^{\ell })^{m}\), we have m subvectors \(x^*_1,\ldots ,x^*_{m}\in \mathbb {R}^{\ell }\) for the optimum \(x^*\). Therefore, if the obtained solutions of the 100 problems are \(x^{*,1},x^{*,2},\ldots ,x^{*,100}\), then we have \(\lambda _i^{\max }:=\max \big \{\lambda _i(x^{*,p}_j)\bigm |(j,p)\in \{1,2,\ldots ,m\}\times \{1,2,\ldots ,100\}\big \}\) and \(\lambda _i^{\min }:=\min \big \{\lambda _i(x^{*,p}_j)\bigm |(j,p)\in \{1,2,\ldots ,m\}\times \{1,2,\ldots ,100\}\big \}\) for each \(i=1,2\).

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Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Shunsuke Hayashi

  2. Department of Mathematics, National Cheng Kung University, Tainan, 70101, Taiwan

    Soon-Yi Wu

  3. National Center for Theoretical Sciences, Tainan, Taiwan

    Soon-Yi Wu

  4. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Liping Zhang

Authors
  1. Shunsuke Hayashi
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  2. Soon-Yi Wu
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  3. Liping Zhang
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Correspondence to Soon-Yi Wu.

Additional information

This research was supported in part by JSPS KAKENHI Grant Number 26330022, and by National Center for Theoretical Sciences, Taiwan.

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Hayashi, S., Wu, SY. & Zhang, L. Computation Algorithm for Convex Semi-infinite Program with Second-Order Cones: Special Analyses for Affine and Quadratic Case. J Sci Comput 68, 573–595 (2016). https://doi.org/10.1007/s10915-015-0149-6

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