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Narrowing the difficulty gap for the Celis–Dennis–Tapia problem

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Abstract

We study the Celis–Dennis–Tapia (CDT) problem: minimize a non-convex quadratic function over the intersection of two ellipsoids. In contrast to the well-studied trust region problem where the feasible set is just one ellipsoid, the CDT problem is not yet fully understood. Our main objective in this paper is to narrow the difficulty gap that occurs when the Hessian of the Lagrangian is indefinite at all Karush–Kuhn–Tucker points. We prove new sufficient and necessary conditions both for local and global optimality, based on copositivity, giving a complete characterization in the degenerate case.

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Notes

  1. [20, Cor.5.4] offers simple sufficient conditions on the problem data \(({\mathsf Q},{\mathsf q},{\mathsf A},{\mathsf a})\) that rule out this phenomenon, namely that no off-diagonal entry of \({\mathsf Q}\) or \({\mathsf A}{\mathsf A}^\top \) is strictly positive and \(\left\{ {\mathsf A}{\mathsf a}, -{\mathsf q}\right\} \subseteq {\mathbb R}^n_+\,\). Note that this particularly implies \({\mathsf H}(\bar{u},\bar{v})\bar{{\mathsf x}}\in {\mathbb R}^n_+\) by (23).

  2. We are indebted to W. Schachinger for this observation (personal communication). Note that when \(n=2\), this means that \({\mathsf H}(\bar{u}(t),\bar{v}(t))\) is \(\Gamma _{\!\!\mathrm red}(\bar{{\mathsf x}})\)-copositive for either \(t=0\) or \(t=1\), but in higher dimensions the relations between the curvatures vary with \({\mathsf d}\), so that whether \(t\) is 0 or 1 in the condition \({\mathsf d}^\top {\mathsf H}(\bar{u}(t),\bar{v}(t)) {\mathsf d}\ge 0\) depends on \({\mathsf d}\).

  3. We used BFGS to minimize the exact penalty function \(p({\mathsf x})=f({\mathsf x})+\rho \max (r({\mathsf x}),0)+\rho \max (s({\mathsf x}),0)\), for some \(\rho >0\) that is increased as needed to ensure feasibility. As discussed in [21], BFGS is a surprisingly effective method to find local minimizers of nonsmooth, nonconvex functions such as \(p\). Since one can expect only to find local minimizers in general, we did this repeatedly from 10 randomly generated starting points for each problem instance, selecting the result \(\bar{{\mathsf x}}\) with the lowest value of \(f\) as our candidate for the global minimizer. The results summarized in Table 1 and discussed below show that in by far the majority of cases, global optimality was confirmed, and in all except one of 70,000 tests at least local optimality was confirmed. (At the suggestion of M. Fampa, we also conducted a similar experiment using up to \(2^n\) starting points instead of 10, stopping as soon as global optimality was confirmed; the outcome was similar.)

  4. The case \(n=2\) receives particular attention in the recent theoretical study [29].

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Acknowledgments

The first author is grateful to Cambridge University and to New York University for the hospitality he enjoyed during his research stays as a visiting fellow. The second author was supported in part by the U.S. National Science Foundation under grant DMS-1317205. Both authors are indebted to an anonymous referee and the Managing Guest Editor for their diligence and helpful constructive remarks.

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Authors and Affiliations

  1. ISOR, University of Vienna, Vienna, Austria

    Immanuel M. Bomze

  2. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Michael L. Overton

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  1. Immanuel M. Bomze
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Correspondence to Immanuel M. Bomze.

Additional information

Dedicated to John E. Dennis Jr. and Richard A. Tapia on the occasion of their 75th birthdays.

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Bomze, I.M., Overton, M.L. Narrowing the difficulty gap for the Celis–Dennis–Tapia problem. Math. Program. 151, 459–476 (2015). https://doi.org/10.1007/s10107-014-0836-3

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