Showing papers by "Tamás Terlaky published in 2019"
TL;DR: In this article, the authors study parametric analysis of semidefinite optimization problems w.r.t. the perturbation of the objective function, and study the behavior of the optimal partition and opti...
Abstract: In this paper, we study parametric analysis of semidefinite optimization problems w.r.t. the perturbation of the objective function. We study the behaviour of the optimal partition and opti...
13 citations
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TL;DR: This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction, and presents a methodology, stemming from numerical algebraic geometry, to efficiently compute nonlinearity intervals and transition points of the optimal partition.
Abstract: This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction. We review the notions of invariancy set, nonlinearity interval, and transition point of the optimal partition, and we investigate their characterizations. We show that the set of transition points is finite and the continuity of the optimal set mapping, on the basis of Painleve-Kuratowski set convergence, might fail on a nonlinearity interval. Under a local nonsingularity condition, we then develop a methodology, stemming from numerical algebraic geometry, to efficiently compute nonlinearity intervals and transition points of the optimal partition. Finally, we support the theoretical results by applying our procedure to some numerical examples.
10 citations
TL;DR: This work uses the approximation of the optimal partition in a rounding procedure to generate an approximate maximally complementary solution from an interior solution, sufficiently close to the optimal set.
8 citations
TL;DR: Under primal and dual nondegeneracy conditions, the quadratic convergence of Newton's method is established to the unique optimal solution of second-order conic optimization, which depends on the optimal partition of the problem, which can be identified from a bounded sequence of interior solutions.
Abstract: Under primal and dual nondegeneracy conditions, we establish the quadratic convergence of Newton's method to the unique optimal solution of second-order conic optimization. Only very few approaches...
6 citations
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TL;DR: Under primal and dual nondegeneracy conditions, it is shown that a transition point can be numerically identified from the higher-order derivatives of the Lagrange multipliers associated with a nonlinear reformulation of the parametric second-order conic optimization problem.
Abstract: In this paper, using an optimal partition approach, we study the parametric analysis of a second-order conic optimization problem, where the objective function is perturbed along a fixed direction. We introduce the notions of nonlinearity interval and transition point of the optimal partition, and we prove that the set of transition points is finite. Additionally, on the basis of Painleve-Kuratowski set convergence, we provide sufficient conditions for the existence of a nonlinearity interval, and we show that the continuity of the primal or dual optimal set mapping might fail on a nonlinearity interval. We then propose, under the strict complementarity condition, an iterative procedure to compute a nonlinearity interval of the optimal partition. Furthermore, under primal and dual nondegeneracy conditions, we show that a transition point can be numerically identified from the higher-order derivatives of the Lagrange multipliers associated with a nonlinear reformulation of the parametric second-order conic optimization problem. Our theoretical results are supported by numerical experiments.
3 citations
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TL;DR: It is shown that a boundary point of a nonlinearity interval can be numerically identified from a non linear reformulation of the parametric second-order conic optimization problem.
Abstract: In this paper, using an optimal partition approach, we study the parametric analysis of a second-order conic optimization problem, where the objective function is perturbed along a fixed direction. We characterize the notions of so-called invariancy set and nonlinearity interval, which serve as stability regions of the optimal partition. We then propose, under the strict complementarity condition, an iterative procedure to compute a nonlinearity interval of the optimal partition. Furthermore, under primal and dual nondegeneracy conditions, we show that a boundary point of a nonlinearity interval can be numerically identified from a nonlinear reformulation of the parametric second-order conic optimization problem. Our theoretical results are supported by numerical experiments.
2 citations
23 Aug 2019
TL;DR: This project uses novel Mixed-Integer Conic Optimization techniques to design aircraft wing structures that take full advantage of new composite materials and manufacturing restrictions to solve discrete truss topology and ply-angle problems.
Abstract: The project's aim is to use novel Mixed-Integer Conic Optimization (MICO) techniques to design aircraft wing structures that take full advantage of new composite materials and manufacturing restrictions. Considering the freedom provided by these design options is key to develop cutting-edge wing designs. However, obtaining (guaranteed) near-optimal solutions for the resulting large-scale structural design problem is very challenging for current solution approaches. The MICO solution approach allows to capture this complexity and obtain near-optimal solutions. In particular, as a result of this project, novel results have been obtained and published regarding the solution of discrete truss topology and ply-angle problems. Distribution Statement DISTRIBUTION A: Distribution approved for public release. This is block 12 on the SF298 form. Distribution A Approved for Public Release Explanation for Distribution Statement If this is not approved for public release, please provide a short explanation. E.g., contains proprietary information. SF298 Form Please attach your SF298 form. A blank SF298 can be found here. Please do not password protect or secure the PDF The maximum file size for an SF298 is 50MB. Terlaky_SF298_print.pdf Upload the Report Document. File must be a PDF. Please do not password protect or secure the PDF . The maximum file size for the Report Document is 50MB. Terlaky_AFOSR_Final_Report.pdf Upload a Report Document, if any. The maximum file size for the Report Document is 50MB. Archival Publications (published) during reporting period: See pages 7 and 8 or the Report. New discoveries, inventions, or patent disclosures: Do you have any discoveries, inventions, or patent disclosures to report for this period? No Please describe and include any notable dates Do you plan to pursue a claim for personal or organizational intellectual property? Changes in research objectives (if any): N/A Change in AFOSR Program Officer, if any: N/A Extensions granted or milestones slipped, if any: One year extension granted in 2018 till June 14, 2019.
TL;DR: A perceptron example is presented to show that ρ may decrease after one rescaling step in this deterministic rescaling method, and it is shown that no complexity proof for the rescaling von Neumann algorithm can be based on increasing ρ monotonically.
Abstract: Recently, Pena and Soheili presented a deterministic rescaling perceptron algorithm and proved that it solves a feasible perceptron problem in O(m2n2 log (ρ−1)) perceptron update steps, whe...