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Local reduction based SQP-type method for semi-infinite programs with an infinite number of second-order cone constraints

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Abstract

The second-order cone program (SOCP) is an optimization problem with second-order cone (SOC) constraints and has achieved notable developments in the last decade. The classical semi-infinite program (SIP) is represented with infinitely many inequality constraints, and has been studied extensively so far. In this paper, we consider the SIP with infinitely many SOC constraints, called the SISOCP for short. Compared with the standard SIP and SOCP, the studies on the SISOCP are scarce, even though it has important applications such as Chebychev approximation for vector-valued functions. For solving the SISOCP, we develop an algorithm that combines a local reduction method with an SQP-type method. In this method, we reduce the SISOCP to an SOCP with finitely many SOC constraints by means of implicit functions and apply an SQP-type method to the latter problem. We study the global and local convergence properties of the proposed algorithm. Finally, we observe the effectiveness of the algorithm through some numerical experiments.

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Notes

  1. For two sets \(X\subseteq Y\), the distance from \(X\) to \(Y\) is defined as \({\mathop {\mathrm{dist}\,}}(X,Y)\,{:=}\,\sup _{y\in Y}{\inf }_{x\in X }{\Vert }x-y{\Vert }\).

  2. This can be regarded as a kind of Cottle’s constraint qualification for \(\mathrm{SOCP}(x^{*},\varepsilon )\).

  3. The fact that \(u^{\top }v=0, u\in {\mathop {\mathrm{int}}}\,\mathcal{K }^m, v\in \mathcal{K }^m \Rightarrow v=0\) is used here.

  4. This spectral decomposition process may be expensive computationally, when the size of \(B_k\) becomes huge. In that case, it may be reasonable to use the modified Cholesky factorization [5] to make \(B_k\) positive-definite.

  5. The origin \(x=0\) always lies in the interior of the feasible region, since we have \(-b^s(t)\in {\mathop {\mathrm{int}}}\, \mathcal{K }^{m_s}\) from \(-b^s_1(t)-\Vert (-b^s_2(t),\ldots ,-b^s_{m_s}(t))^{\top }\Vert >0\) for all \(t\in T\).

  6. We also suppose that Assumption 4.1(b) holds.

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Acknowledgments

We would like to thank two anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Management Science, Faculty of Engineering, Division I, Tokyo University of Science, Tokyo,  162-8601, Japan

    Takayuki Okuno

  2. Department of Information Systems and Mathematical Science, Faculty of Information System and Engineering, Nanzan University, Aichi,  489-0863, Japan

    Masao Fukushima

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  1. Takayuki Okuno
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  2. Masao Fukushima
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Correspondence to Masao Fukushima.

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This research was supported in part by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science.

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Okuno, T., Fukushima, M. Local reduction based SQP-type method for semi-infinite programs with an infinite number of second-order cone constraints. J Glob Optim 60, 25–48 (2014). https://doi.org/10.1007/s10898-013-0063-0

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