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Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2023年第2期86卷 555-598页

作者： Arahata, Shun Okuno, Takayuki Takeda, Akiko Univ Tokyo Grad Sch Informat Sci & Technol Tokyo Japan RIKEN Ctr Adv Intelligence Project Tokyo Japan Seikei Univ Fac Sci & Technol Dept Sci & Technol Tokyo Japan

We propose a primal-dual interior-point method (IPM) with convergence to second-order stationary points (SOSPs) of nonlinear semidefinite optimization problems, abbreviated as NSDPs. As far as we know, the current algorithms for NSDPs only ensure convergence to first-order stationary points such as Karush-Kuhn-Tucker points, but without a worst-case iteration complexity. The proposed method generates a sequence approximating SOSPs while minimizing a primal-dual merit function for NSDPs by using scaled gradient directions and directions of negative curvature. Under some assumptions, the generated sequence accumulates at an SOSP with a worst-case iteration complexity. This result is also obtained for a primal IPM with a slight modification. Finally, our numerical experiments show the benefits of using directions of negative curvature in the proposed method.

关键词： nonlinear semidefinite programming Primal-dual interior-point method Negative curvature direction Second-order stationary points

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Robust State Estimation for Linear Systems Under Distributional Uncertainty

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2021年 69卷 5963-5978页

作者： Wang, Shixiong Wu, Zhongming Lim, Andrew Natl Univ Singapore Dept Ind Syst Engn & Management Singapore 117576 Singapore Nanjing Univ Informat Sci & Technol Sch Management Sci & Engn Nanjing 210044 Peoples R China Southwest Jiaotong Univ Sch Comp & Artificial Intelligence Chengdu 611756 Peoples R China Red Jasper Holdings Singapore 128424 Singapore

Modeling uncertainties for real linear systems are unavoidable. These uncertainties can significantly degrade the performance of optimal state estimators designed for nominal system models. The challenge is quantifying such uncertainties and devising robust estimators that are insensitive to them. This paper is therefore concerned with distributionally robust state estimation for linear Markov systems. We propose a new modeling framework that describes uncertainties using a family of distributions so that the worst-case robust estimate in the state space is made over the least-favorable distribution. This framework uses only one or two scalars to express the uncertainty set and does not require the structural information of model uncertainties. Furthermore, the moment-based ambiguity set is suggested to embody the distributional uncertainty family. As a result, the estimation problem transforms into a nonlinear semidefinite program with linear constraints, which can be analytically and efficiently solved. Intensive experiments illustrate the advantages of the proposed framework over existing methods.

关键词： Uncertainty Linear systems State estimation Estimation Measurement uncertainty Optimization Kalman filters State estimation linear system distributional robustness model uncertainty moment ambiguity set nonlinear semidefinite programming

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Local and superlinear convergence of a primal-dual interior point method for nonlinear semidefinite programming

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MATHEMATICAL programming 2012年第1-2期132卷 1-30页

作者： Yamashita, Hiroshi Yabe, Hiroshi Math Syst Inc Shinjuku Ku Tokyo 1600022 Japan Tokyo Univ Sci Dept Math Informat Sci Fac Sci Shinjuku Ku Tokyo 1628601 Japan

In this paper, we consider a primal-dual interior point method for solving nonlinear semidefinite programming problems. We propose primal-dual interior point methods based on the unscaled and scaled Newton methods, which correspond to the AHO, HRVW/KSH/M and NT search directions in linear SDP problems. We analyze local behavior of our proposed methods and show their local and superlinear convergence properties.

关键词： nonlinear semidefinite programming Primal-dual interior point method Local and superlinear convergence

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A primal-dual interior point method for nonlinear semidefinite programming

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MATHEMATICAL programming 2012年第1-2期135卷 89-121页

作者： Yamashita, Hiroshi Yabe, Hiroshi Harada, Kouhei Tokyo Univ Sci Fac Sci Dept Math Informat Sci Shinjuku Ku Tokyo 1628601 Japan Math Syst Inc Shinjuku Ku Tokyo 1600022 Japan

This paper is concerned with a primal-dual interior point method for solving nonlinear semidefinite programming problems. The method consists of the outer iteration (SDPIP) that finds a KKT point and the inner iteration (SDPLS) that calculates an approximate barrier KKT point. Algorithm SDPLS uses a commutative class of Newton-like directions for the generation of line search directions. By combining the primal barrier penalty function and the primal-dual barrier function, a new primal-dual merit function is proposed. We prove the global convergence property of our method. Finally some numerical experiments are given.

关键词： nonlinear semidefinite programming Primal-dual interior point method Barrier penalty function Primal-dual merit function Global convergence

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Free material optimization with local stress constraints for laminated structures

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ENGINEERING OPTIMIZATION 2020年第4期52卷 567-582页

作者： Weldeyesus, Alemseged Gebrehiwot Univ Edinburgh Sch Math Edinburgh Midlothian Scotland Univ Edinburgh Maxwell Inst Math Sci Edinburgh Midlothian Scotland

In Free Material Optimization (FMO), the design variable is the elastic material tensor, which is allowed to change its values freely over the design domain resulting in both optimal material properties and optimal material distribution. Models are mostly available for two- and three-dimensional continuum structures. Recently, these models have been extended to laminated plates and shells. The goal of this article is to introduce constraints on local stresses to these new models. The associated optimization problems are highly nonlinear semidefinite programming problems that are known to be challenging for numerical tractability, and are solved by a primal-dual interior point method previously proposed for FMO problems without stress constraints. The algorithm utilizes the FMO problem structure, wherein the matrix inequality constraints involve small many block matrices. Several numerical experiments demonstrate the stress constrained models and the capability of the method to obtain solutions to these problems within a modest number of iterations.

关键词： Structural optimization free material optimization nonlinear semidefinite programming stress constraints

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Truss geometry and topology optimization with global stability constraints

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2020年第4期62卷 1721-1737页

作者： Weldeyesus, Alemseged Gebrehiwot Gondzio, Jacek He, Linwei Gilbert, Matthew Shepherd, Paul Tyas, Andrew Univ Edinburgh Sch Math Peter Guthrie Tait Rd Edinburgh EH9 3FD Midlothian Scotland Univ Edinburgh Maxwell Inst Math Sci Peter Guthrie Tait Rd Edinburgh EH9 3FD Midlothian Scotland NASK Res Inst Kolska 12 PL-01045 Warsaw Poland Univ Sheffield Dept Civil & Struct Engn Sir Frederick Mappin BldgMappin St Sheffield S1 3JD S Yorkshire England Univ Bath Dept Architecture & Civil Engn Bath BA2 7AY Avon England

In this paper, we introduce geometry optimization into an existing topology optimization workflow for truss structures with global stability constraints, assuming a linear buckling analysis. The design variables are the cross-sectional areas of the bars and the coordinates of the joints. This makes the optimization problem formulations highly nonlinear and yields nonconvex semidefinite programming problems, for which there are limited available numerical solvers compared with other classes of optimization problems. We present problem instances of truss geometry and topology optimization with global stability constraints solved using a standard primal-dual interior point implementation. During the solution process, both the cross-sectional areas of the bars and the coordinates of the joints are concurrently optimized. Additionally, we apply adaptive optimization techniques to allow the joints to navigate larger move limits and to improve the quality of the optimal designs.

关键词： Geometry and topology optimization Global stability nonlinear semidefinite programming Interior point methods

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Including global stability in truss layout optimization for the conceptual design of large-scale applications

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2018年第3期57卷 1213-1232页

作者： Tugilimana, Alexis Coelho, Rajan Filomeno Thrall, Ashley P. Univ Libre Bruxelles BATir Dept Ave F Roosevelt50CP 194-2 B-1050 Brussels Belgium Univ Notre Dame Dept Civil & Environm Earth Sci 159 Fitzpatrick Hall Notre Dame IN 46556 USA

Including stability in truss topology optimization is critical to avoid unstable optimized designs in practical applications. While prior research addresses this challenge by implementing local buckling and linear prebuckling, numerical difficulties remain due to the global stability singularity phenomenon. Therefore, the goal of this paper is to develop an optimization formulation for truss topology optimization including global stability without numerical singularities, within the framework of the preliminary design of large-scale structures. This task is performed by considering an appropriate simultaneous analysis and design formulation, in which the use of a disaggregated form for the equilibrium equations alleviates the singularities inherent to global stability. By implementing a local buckling criterion for hollow truss elements, the resulting formulation is well-suited for the preliminary design of large-scale trusses in civil engineering applications. Three applications illustrate the efficiency of the proposed approach, including a benchmark truss structure and the preliminary design of a footbridge and a dome. The results demonstrate that including local buckling and global stability can considerably affect the optimized design, while offering a systematic means of avoiding unstable solutions. It is also shown that the proposed approach is in a good agreement with linear prebuckling assumptions.

关键词： Truss topology optimization Local buckling Global stability nonlinear semidefinite programming Simultaneous analysis and design

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Existence of augmented Lagrange multipliers: reduction to exact penalty functions and localization principle

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MATHEMATICAL programming 2017年第1-2期166卷 297-326页

作者： Dolgopolik, M. V. St Petersburg State Univ Univ Skii Ave 35 St Petersburg 198504 Russia

In this article, we present new general results on existence of augmented Lagrange multipliers. We define a penalty function associated with an augmented Lagrangian, and prove that, under a certain growth assumption on the augmenting function, an augmented Lagrange multiplier exists if and only if this penalty function is exact. We also develop a new general approach to the study of augmented Lagrange multipliers called the localization principle. The localization principle allows one to study the local behaviour of the augmented Lagrangian near global optimal solutions of the initial optimization problem in order to prove the existence of augmented Lagrange multipliers.

关键词： Augmented Lagrangian Lagrange multipliers Exact penalty function Localization principle nonlinear semidefinite programming

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AN EQUIVALENCY CONDITION OF NONSINGULARITY IN nonlinear semidefinite programming

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Journal of Systems Science & Complexity 2010年第4期23卷 822-829页

作者： Chengjin LI Wenyu SUN Raimundo J. B. de SAMPAIO School of Mathematical Sciences Nanjing Normal and Computer Science Fujian Normal University Wenyu SUN University Nanjing 210046 China School of Mathematics Fuzhou 350007 China School of Mathematical Sciences Nanjing Normal University Nanjing 210046 China Pontifical Catholic University of Parana ( PUCPR) Graduate Program in Production and Systems Engineering (PPGEPS) CEP 81611-970 Curitiba Parana Brazil

In this paper,an equivalency condition of nonsingularity in nonlinear semidefinite programming,which can be viewed as a generalization of the equivalency condition of nonsingularity for linearsemidefinite programming,... 详细信息

关键词： Equivalency condition of nonsingularity nonlinear semidefinite programming semidefinite programming.

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Sufficient optimality conditions hold for almost all nonlinear semidefinite programs

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MATHEMATICAL programming 2016年第1-2期158卷 77-97页

作者： Dorsch, Dominik Gomez, Walter Shikhman, Vladimir Rhein Westfal TH Aachen Chair Math Anal C D-52056 Aachen Germany Univ La Frontera Dept Ingn Matemat Ave Francisco Salazar Temuco 01145 Chile UCL CORE B-1348 Louvain La Neuve Belgium

We derive a new genericity result for nonlinear semidefinite programming (NLSDP). Namely, almost all linear perturbations of a given NLSDP are shown to be nondegenerate. Here, nondegeneracy for NLSDP refers to the transversality constraint qualification, strict complementarity and second-order sufficient condition. Due to the presence of the second-order sufficient condition, our result is a nontrivial extension of the corresponding results for linear semidefinite programs (SDP) from Alizadeh et al. (Math Program 77(2, Ser. B):111-128, 1997). The proof of the genericity result makes use of Forsgren's derivation of optimality conditions for NLSDP in Forsgren (Math Program 88(1, Ser. A):105-128, 2000). Due to the latter approach, the positive semidefiniteness of a symmetric matrix G(x), depending continuously on x, is locally equivalent to the fact that a certain Schur complement S(x) of G(x) is positive semidefinite. This yields a reduced NLSDP by considering the new semidefinite constraint , instead of . While deriving optimality conditions for the reduced NLSDP, the well-known and often mentioned "H-term" in the second-order sufficient condition vanishes. This allows us to access the proof of the genericity result for NLSDP.

关键词： nonlinear semidefinite programming Schur complement H-term Genericity

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