Abstract
Optimality functions pioneered by E. Polak characterize stationary points, quantify the degree with which a point fails to be stationary, and play central roles in algorithm development. For optimization problems requiring approximations, optimality functions can be used to ensure consistency in approximations, with the consequence that optimal and stationary points of the approximate problems indeed are approximately optimal and stationary for an original problem. In this paper, we review the framework and illustrate its application to nonlinear programming and other areas. Moreover, we introduce lopsided convergence of bifunctions on metric spaces and show that this notion of convergence is instrumental in establishing consistency of approximations. Lopsided convergence also leads to further characterizations of stationary points under perturbations and approximations.
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Notes
We recall that the outer limit of a sequence of sets \(\{A^\nu \}_{\nu \in \mathbb {N}}\), denoted by \(\mathop {{\mathrm{limsup}}}\nolimits A^\nu \), is the collection of points y to which a subsequence of \(\{y^\nu \}_{\nu \in \mathbb {N}}\), with \(y^\nu \in A^\nu \), converges. The inner limit, denoted by \(\mathop {{\mathrm{liminf}}}\nolimits A^\nu \), is the points to which a sequence of \(\{y^\nu \}_{\nu \in \mathbb {N}}\), with \(y^\nu \in A^\nu \), converges. If both limits exist and are identical, we say that the set is the Painlevé–Kuratowski limit of \(\{A^\nu \}_{\nu \in \mathbb {N}}\) and that \(A^\nu \) set-converges to this set; see [18, 20].
Here, we consider set-convergence of subsets of \(\mathcal{X}\times \mathbb {R}\), which is equipped with the metric \(\rho ((x,x_0),(x',x_0'))=\max \{d_\mathcal{X}(x,x'),|x_0-x_0'|\}\) for \(x,x'\in \mathcal{X}\) and \(x_0,x_0'\in \mathbb {R}\).
We note that Theorem 3.1 is stated for finite-valued functions and \(h^\nu \) and h might be extended real-valued. However, the conclusions hold under this slight extension.
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Acknowledgments
This material is based upon work supported in part by the US Army Research Laboratory and the US Army Research Office under Grant Numbers 00101-80683, W911NF-10-1-0246 and W911NF-12-1-0273.
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We dedicate this paper to our long-time friend, colleague, collaborator, and advisor Elijah (Lucien) Polak in honor of his outstanding career and to celebrate his 85th birthday. We wish him fair weather and following snow conditions.
Communicated by Igor Konnov.
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Royset, J.O., Wets, R.JB. Optimality Functions and Lopsided Convergence. J Optim Theory Appl 169, 965–983 (2016). https://doi.org/10.1007/s10957-015-0839-0
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DOI: https://doi.org/10.1007/s10957-015-0839-0
Keywords
- epi-Convergence
- Lopsided convergence
- Consistent approximations
- Optimality functions
- Optimality conditions