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Extension of the LP-Newton method to conic programming problems via semi-infinite representation

NUMERICAL ALGORITHMS 2021年第3期86卷 1285-1302页

作者： Tanaka, Mirai Okuno, Takayuki Inst Stat Math Dept Stat Inference & Math 10-3 Midori Cho Tachikawa Tokyo 1908562 Japan RIKEN Ctr Adv Intelligence Project Chuo Ku 1-4-1 Nihonbashi Tokyo 1030027 Japan

The LP-Newton method solves linear programming (LP) problems by repeatedly projecting a current point onto a certain relevant polytope. In this paper, we extend the algorithmic framework of the LP-Newton method to conic programming (CP) problems via a linear semi-infinite programming (LSIP) reformulation. In this extension, we produce a sequence by projection onto polyhedral cones constructed from LP problems obtained by finitely relaxing the LSIP problem equivalent to the CP problem. We show global convergence of the proposed algorithm under mild assumptions. To investigate its efficiency, we apply our proposed algorithm and a primal-dual interior-point method to second-order cone programming problems and compare the obtained results.

关键词： conic programming Semi-infinite programming Adaptive polyhedral approximation LP-Newton method

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Optimality conditions and total dualities for conic programming involving composite function

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OPTIMIZATION 2020年第2期69卷 305-327页

作者： Fang, Donghui Zhang, Yong Jishou Univ Coll Math & Stat Jishou Peoples R China

The paper concerns the study of a new class of conic programming with objectives given as the difference of a composite function and a convex function. We first introduce two new notions of regularity conditions in terms of the subdifferential of the involving functions. Under the new regularity conditions, we provide some necessary and/or sufficient conditions for KKT type optimality conditions to hold. Similarly, saddle point theorems and total Lagrange dualities for conic programming are also given.

关键词： Optimality condition total duality saddle point theorem composite function conic programming

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Extended Farkas's Lemmas and Strong Dualities for conic programming Involving Composite Functions

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2018年第2期176卷 351-376页

作者： Fang, D. H. Zhang, Y. Jishou Univ Coll Math & Stat Jishou 416000 Peoples R China

The paper is devoted to the study of a new class of conic constrained optimization problems with objectives given as differences of a composite function and a convex function. We first introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions. Under the new constraint qualifications, we provide necessary and sufficient conditions for several versions of Farkas lemmas to hold. Similarly, we provide characterizations for conic constrained optimization problems to have the strong or stable strong dualities such as Lagrange, Fenchel-Lagrange or Toland-Fenchel-Lagrange duality.

关键词： Farkas lemma Strong duality Composite functions Constraint qualifications conic programming

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Linear and conic programming estimators in high dimensional errors-in-variables models

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JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY 2017年第3期79卷 939-956页

作者： Belloni, Alexandre Rosenbaum, Mathieu Tsybakov, Alexandre B. Duke Univ Durham NC 27706 USA Univ Paris 06 Paris France Ctr Rech Econ & Stat Paris France

We consider the linear regression model with observation error in the design. In this setting, we allow the number of covariates to be much larger than the sample size. Several new estimation methods have been recently introduced for this model. Indeed, the standard lasso estimator or Dantzig selector turns out to become unreliable when only noisy regressors are available, which is quite common in practice. In this work, we propose and analyse a new estimator for the errors-in-variables model. Under suitable sparsity assumptions, we show that this estimator attains the minimax efficiency bound. Importantly, this estimator can be written as a second-order cone programming minimization problem which can be solved numerically in polynomial time. Finally, we show that the procedure introduced by Rosenbaum and Tsybakov, which is almost optimal in a minimax sense, can be efficiently computed by a single linear programming problem despite non-convexities.

关键词： conic programming Dantzig selector Linear regression model Matrix uncertainty selector Prediction sensitivity

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Strong calmness of perturbed KKT system for a class of conic programming with degenerate solutions

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OPTIMIZATION 2019年第6期68卷 1131-1156页

作者： Liu, Y. L. Pan, S. H. Guangdong Univ Technol Sch Appl Math Guangzhou Guangdong Peoples R China South China Univ Technol Sch Math Guangzhou Guangdong Peoples R China

This paper is concerned with the strong calmness of the KKT solution mapping for a class of canonically perturbed conic programming, which plays a central role in achieving fast convergence under situations when the Lagrange multiplier associated to a solution of these conic optimization problems is not unique. We show that the strong calmness of the KKT solution mapping is equivalent to a local error bound for the solutions to the perturbed KKT system, and is also equivalent to the pseudo-isolated calmness of the stationary point mapping along with the calmness of the multiplier set mapping at the corresponding reference point. Sufficient conditions are also provided for the strong calmness by establishing the pseudo-isolated calmness of the stationary point mapping in terms of the noncriticality of the associated multiplier, and the calmness of the multiplier set mapping in terms of a relative interior condition for the multiplier set. These results cover and extend the existing ones in Hager and Gowda [Stability in the presence of degeneracy and error estimation. Math Program. 1999;85:181-192];Izmailov and Solodov [Stabilized SQP revisited. Math Program. 2012;133:93-120] for nonlinear programming and in Cui et al. [On the asymptotic superlinear convergence of the augmented Lagrangian method for semidefinite programming with multiple solutions. 2016, arXiv: 1610.00875v1];Zhang and Zhang [Critical multipliers in semidefinite programming. 2018, arXiv: 1801.02218v1] for semidefinite programming.

关键词： conic programming KKT solution mapping strong calmness local error bound pseudo-isolated calmness noncritical multiplier

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Square-root lasso: pivotal recovery of sparse signals via conic programming

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BIOMETRIKA 2011年第4期98卷 791-806页

作者： Belloni, A. Chernozhukov, V. Wang, L. Duke Univ Fuqua Sch Business Durham NC 27708 USA MIT Dept Econ Cambridge MA 02142 USA MIT Dept Math Cambridge MA 02139 USA

We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation Sigma nor does it need to pre-estimate Sigma. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate Sigma{(s/n) log p}(1/2) in the prediction norm, and thus matching the performance of the lasso with known Sigma. These performance results are valid for both Gaussian and non-Gaussian errors, under some mild moment restrictions. We formulate the square-root lasso as a solution to a convex conic programming problem, which allows us to implement the estimator using efficient algorithmic methods, such as interior-point and first-order methods.

关键词： conic programming High-dimensional sparse model Moderate deviation theory

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Cascading: an adjusted exchange method for robust conic programming

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CENTRAL EUROPEAN JOURNAL OF OPERATIONS RESEARCH 2008年第2期16卷 179-189页

作者： Werner, Ralf Hypo Real Estate Holding AG Grp Risk Control D-80538 Munich Germany

It is well known that the robust counterpart introduced by Ben-Tal and Nemirovski (Math Oper Res 23:769-805, 1998) increases the numerical complexity of the solution compared to the original problem. Kocvara, Nemirovski and Zowe therefore introduced in Kocvara et al. (Comput Struct 76:431-442, 2000) an approximation algorithm for the special case of robust material optimization, called cascading. As the title already indicates, we will show that their method can be seen as an adjustment of standard exchange methods to semi-infinite conic programming. We will see that the adjustment can be motivated by a suitable reformulation of the robust conic problem.

关键词： robust programming conic programming semi-infinite programming exchange method

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A Hybrid Optimization Algorithm for Distribution Network Coordinated Operation with SNOP Based on Simulated Annealing and conic programming

A Hybrid Optimization Algorithm for Distribution Network Coo...

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IEEE-Power-and-Energy-Society General Meeting (PESGM)

作者： Dong, Xuzhu Wu, Zhengrong Song, Guanyu Ji, Haoran Li, Peng Wang, Chengshan China Southern Power Grid Smart Grid Res Inst Elect Power Res Inst Guangzhou Guangdong Peoples R China Tianjin Univ Minist Educ Key Lab Smart Grid Tianjin Peoples R China

ISBN: (纸本)9781509041688

Soft normally open point (SNOP) is a novel power electronic device installed in place of normally-open point. The application of SNOP will greatly promote the flexibility and controllability of distribution network. Considering the high investment of SNOP, both tie switch and SNOP should be taken into account in the coordinated operation problem of distribution network. Firstly, the optimization model for distribution network coordinated operation with SNOP and tie switch is proposed. Then, combining the simulated annealing method with conic programming, this paper proposes a hybrid optimization algorithm, involving simulated annealing method to obtain the switch states and conic programming to optimize the transmitted power of SNOP. This hybrid algorithm can solve the above large-scale mixed-integer nonlinear problem accurately and rapidly, while satisfying the demand of distribution network coordinated operation. Finally, the IEEE 69-node system is used to demonstrate the effectiveness of the proposed hybrid algorithm.

关键词： simulated annealing conic programming soft normally open point (SNOP) network reconfiguration

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A conic programming Approach to Optimal Transmission Switching Considering Reactive Power and Voltage Security

A Conic Programming Approach to Optimal Transmission Switchi...

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General Meeting of the IEEE-Power-and-Energy-Society

作者： Bai, Yang Zhong, Haiwang Xia, Qing Wang, Yang Tsinghua Univ Dept Elect Engn Beijing Peoples R China North China Power Dispatching & Control Branch Ct Beijing Peoples R China

ISBN: (纸本)9781467380409

Optimal transmission switching (OTS) exploits the flexibility in grid topology to reduce the system dispatch cost. However, DC-power-flow-based OTS solution cannot guarantee a feasible AC dispatch, which is one of the challenges in practical implementation. To bridge the gap between the theoretical basis and practical implementation of OTS, this paper proposes a conic programming approach to the OTS problem. A mixed integer second order cone programming model is formulated to allow for incorporating reactive power and voltage security constraints, which significantly improves the AC feasibility of the OTS solution. The efficacy of the proposed model is illustrated in the IEEE 57-bus system.

关键词： Optimal transmission switching (OTS) conic programming mixed integer second order cone programming (MISOCP) voltage security

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Cascading: an adjusted exchange method for robust conic programming

Cascading: an adjusted exchange method for robust conic prog...

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5th Europe Workshop on Advances in Continuous Optimization

作者： Werner, Ralf Hypo Real Estate Holding AG Grp Risk Control D-80538 Munich Germany

关键词： robust programming conic programming semi-infinite programming exchange method

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