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A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2010年第3期207卷 1210-1220页

作者： Sun, Jie Zhang, Su Natl Univ Singapore Dept Decis Sci Singapore 119245 Singapore Natl Univ Singapore Risk Management Inst Singapore 119245 Singapore Nankai Univ Sch Business Inst Modern Management Tianjin Peoples R China

We propose a modified alternating direction method for solving convex quadratically constrained quadratic semidefinite optimization problems. The method is a first-order method, therefore requires much less computational effort per iteration than the second-order approaches such as the interior point methods or the smoothing Newton methods. In fact, only a single inexact metric projection onto the positive semidefinite cone is required at each iteration. We prove global convergence and provide numerical evidence to show the effectiveness of this method. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Alternating direction method conic programming Quadratic semidefinite optimization

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Regularity conditions characterizing Fenchel-Lagrange duality and Farkas-type results in DC infinite programming

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2014年第2期414卷 590-611页

作者： Sun, Xiang-Kai Chongqing Technol & Business Univ Coll Math & Stat Chongqing 400067 Peoples R China

In this paper, we consider a DC infinite programming problem (P) with inequality constraints. By using the properties of the epigraph of the conjugate functions, we introduce some new notions of regularity conditions for (P). Under these new regularity conditions, we completely characterize the Fenchel-Lagrange duality and the stable Fenchel-Lagrange duality for (P). Similarly, we also completely characterize the Farkas-type results and the stable Farkas-type results for (P). As applications, we obtain the corresponding results for conic programming problems. (C) 2014 Elsevier Inc. All rights reserved.

关键词： DC infinite programming Regularity conditions Fenchel-Lagrange duality Farkas-type results conic programming

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TRACTABLE ROBUST EXPECTED UTILITY AND RISK MODELS FOR PORTFOLIO OPTIMIZATION

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MATHEMATICAL FINANCE 2010年第4期20卷 695-731页

作者： Natarajan, Karthik Sim, Melvyn Uichanco, Joline Natl Univ Singapore Dept Math NUS Risk Management Inst Singapore Singapore MIT Cambridge MA 02139 USA

Expected utility models in portfolio optimization are based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance, and support information. No additional restrictions on the type of distribution such as normality is made. The investor's utility is modeled as a piecewise-linear concave function. We derive exact and approximate optimal trading strategies for a robust (maximin) expected utility model, where the investor maximizes his worst-case expected utility over a set of ambiguous distributions. The optimal portfolios are identified using a tractable conic programming approach. Extensions of the model to capture asymmetry using partitioned statistics information and box-type uncertainty in the mean and covariance matrix are provided. Using the optimized certainty equivalent framework, we provide connections of our results with robust or ambiguous convex risk measures, in which the investor minimizes his worst-case risk under distributional ambiguity. New closed-form results for the worst-case optimized certainty equivalent risk measures and optimal portfolios are provided for two- and three-piece utility functions. For more complicated utility functions, computational experiments indicate that such robust approaches can provide good trading strategies in financial markets.

关键词： robust portfolio optimization expected utility ambiguity conic programming

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Primal interior-point decomposition algorithms for two-stage stochastic extended second-order cone programming

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OPTIMIZATION 2018年第12期67卷 2291-2323页

作者： Alzalg, Baha Univ Jordan Dept Math Amman 11942 Jordan

We study and solve the two-stage stochastic extended second-order cone programming problem. We show that the barrier recourse functions and the composite barrier functions for this optimization problem are self-concordant families with respect to barrier parameters. These results are used to develop primal decomposition-based interior-point algorithms. The worst case iteration complexity of the developed algorithms is shown to be the same as that for the short- and long-step primal interior algorithms applied to the extensive formulation of our problem.

关键词： Extended second order cone conic programming stochastic programming interior-point algorithms self-concordance

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Safely Learning Dynamical Systems from Short Trajectories 3

Safely Learning Dynamical Systems from Short Trajectories

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3rd Annual Conference on Learning for Dynamics and Control (L4DC)

作者： Ahmadi, Amir Ali Chaudhry, Abraar Sindhwani, Vikas Tu, Stephen Princeton Univ ORFE Princeton NJ 08544 USA Google Robot New York NY USA

A fundamental challenge in learning to control an unknown dynamical system is to reduce model uncertainty by making measurements while maintaining safety. In this work, we formulate a mathematical definition of what it means to safely learn a dynamical system by sequentially deciding where to initialize the next trajectory. In our framework, the state of the system is required to stay within a given safety region under the (possibly repeated) action of all dynamical systems that are consistent with the information gathered so far. For our first two results, we consider the setting of safely learning linear dynamics. We present a linear programming-based algorithm that either safely recovers the true dynamics from trajectories of length one, or certifies that safe learning is impossible. We also give an efficient semidefinite representation of the set of initial conditions whose resulting trajectories of length two are guaranteed to stay in the safety region. For our final result, we study the problem of safely learning a nonlinear dynamical system. We give a second-order cone programming based representation of the set of initial conditions that are guaranteed to remain in the safety region after one application of the system dynamics.

关键词： learning dynamical systems safe learning uncertainty quantification robust optimization conic programming

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Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones 25th

Integer Points in Arbitrary Convex Cones: The Case of the PS...

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25th International Conference on Integer programming and Combinatorial Optimization (IPCO)

作者： De Loera, Jesus A. Marsters, Brittney Xu, Luze Zhang, Shixuan Univ Calif Davis Davis CA 95616 USA Texas A&M Univ College Stn TX 77843 USA

ISBN: (纸本)9783031598340;9783031598357

We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a notion of finite generating set. We show this is true for the cone of positive semidefinite matrices (PSD) and the second-order cone (SOC). Both cones have a finite generating set of integer points, similar in spirit to Hilbert bases, under the action of a finitely generated group. We also extend notions of total dual integrality, Gomory-Chv ' atal closure, and Carath ' eodory rank to integer points in arbitrary cones.

关键词： Integer Points Convex Cones Semigroups Hilbert bases conic programming Positive Semidefinite Cone Second-Order Cone

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CAPACITATED ASSORTMENT OPTIMIZATION AND PRICING PROBLEMS UNDER MIXED MULTINOMIAL LOGIT MODEL

CAPACITATED ASSORTMENT OPTIMIZATION AND PRICING PROBLEMS UND...

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作者： Mehdi Ghaniabadi Bilkent University

学位级别：硕士

We study capacitated assortment optimization problem under mixed multino- mial logit model where a retailer wants to choose the set of products to offer to various customer segments with the goal of maximizing revenue while satisfying different capacity constraints. Each customer segment is identified with a unique purchase behaviour modelled by multinomial logit demand. We consider three general cases of capacity constraints: single resource constraint, multiple resource constraints and multiple cardinality constraints. This problem is NP-hard and there exist two approaches to find exact solutions: formulating the problem as a mixed integer linear program (MILP) or a mixed integer conic quadratic program (conic). For each constraint structure, we develop new efficient procedures to derive McCormick valid inequalities. We provide extensive numerical studies the results of which demonstrate that when the conic model is accompanied with the McCormick inequalities, the problem can be solved effectively even for large sized instances using a commercial optimization software. We also study joint pricing and assortment optimization problem with a single cardinality constraint and establish a new procedure to construct McCormick inequalities. We then present the related numerical studies which indicate that the conic formulation accomplishes the best outcome in the presence of the McCormick inequalities.

关键词： Assortment optimization mixed multinomial logit capacity constraints pricing conic programming

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Automated formulation and resolution of limit analysis problems

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COMPUTERS & STRUCTURES 2021年 243卷 106341-106341页

作者： Bleyer, Jeremy Hassen, Ghazi Univ Gustave Eiffel Lab Navier CNRS Ecole Ponts ParisTech 6-8 Av Blaise Pascal F-77455 Champs Sur Marne France

The present manuscript presents a framework for automating the formulation and resolution of limit analysis problems in a very general manner. This framework relies on FEniCS domain-specific language and the representation of material strength criteria and their corresponding support function in the conic programming setting. Various choices of finite element discretization, including discontinuous Galerkin interpolations, are offered by FEniCS, enabling to formulate lower bound equilibrium elements or upper bound elements including discontinuities for instance. The numerical resolution of the corresponding optimization problem is carried out by the interior-point solver Mosek which takes advantage of the conic representation for yield criteria. Through various illustrative examples ranging from classical continuum limit analysis problems to generalized mechanical models such as plates, shells, strain gradient or Cosserat continua, we show that limit analysis problems can be formulated using only a few lines of code, discretized in a very simple manner and solved extremely efficiently. This paper is accompanied by a FEniCS toolbox implementing the above-mentioned framework. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Limit analysis Yield design Convex optimization conic programming FEniCS Generalized continua

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Shadow price based co-ordination methods of microgrids and battery swapping stations

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APPLIED ENERGY 2019年 253卷 113510-000页

作者： Wang, Yang Lai, Kexing Chen, Fengyun Li, Zhengming Hu, Chunhua Jiangsu Univ Sch Elect & Informat Engn Zhenjiang 212013 Jiangsu Peoples R China Ohio State Univ Dept Elect & Comp Engn Columbus OH 43210 USA Zhenjiang Coll Dept Elect & Informat Sci Zhanjiang 212028 Jiangsu Peoples R China

The growing market share of electric vehicles embodies the remarkable progress of transportation electrification. Battery swapping station serves as a critical infrastructure for efficient electric vehicles refueling. To coordinate the scheduling of a battery swapping station and a microgrid managed by non-cooperative entities, we develop two shadow price-based coordination methods, namely peer-to-peer method and leader-follower method. The context is of great importance for mitigating potential barriers of integration of battery swapping stations and microgrids. The peer-to-peer method only requires the exchange of individual shadow price and power trading requests between two entities. This mechanism is the first-of-its-kind solution that allows for co-ordination between two systems without releasing proprietary information. An iterative heuristic algorithm is developed to obtain outcomes of the co-ordination mechanism. For the leader-follower method, microgrid entity gets access to proprietary information of battery swapping station that is managed by an independent entity. A bi-level optimization model is developed to model the gameplay between two non-corporative entities, which is solved by a modified nested column-and-constraint generation algorithm. For both methods, we use an AC optimal power flow model to optimize operation of microgrid while battery swapping station operation is determined by a mixed-integer linear programming model. This work is also the first-of-its-kind study that incorporates AC power flow equations into relevant research. We demonstrate the effectiveness of both methods using an integrated system comprising a standard IEEE 33-bus system and a battery swapping station serving multiple private electric sedans or public electric buses. Nearly 10% of total operational cost saving can be attained if the proposed methods are applied for coordinating two systems, compared to implementing a baseline method.

关键词： Battery swapping station Bi-level optimization conic programming Decomposition Microgrid scheduling Shadow-price

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A Pseudo-Lower Bound Solution of Structural Bearing Capacity by Bezier Extraction-Based Isogeometric Analysis

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INTERNATIONAL JOURNAL OF APPLIED MECHANICS 2023年第8期15卷

作者： Do, Hien V. Ho, Phuc L. H. Le, Canh V. Nguyen-Xuan, H. HCMC Univ Technol & Educ Linh Chieu Ward Thu Duc City Fac Mech Engn 1 Vo Van Ngan St Ho Chi Minh City Vietnam Dong A Univ Univ Core Res Ctr Disaster free & Safe Ocean City Busan 49315 South Korea Int Univ VNU HCM Sch Civil Engn & Management Quarter 6 Linh Trung Ward Ho Chi Minh City Vietnam HUTECH Univ CIRTech Inst 475A Dien Bien Phu Ward 25 Binh Thanh Dist Ho Chi Minh City Vietnam

This study proposes a pseudo-lower bound method for direct limit analysis of two-dimensional structures and safety evaluation based on isogeometric analysis integrated through Bezier extraction. The key idea in this approach is that the stress field is separated into two parts: fictitious elastic and residual, and then the equilibrium conditions are recast by the weak form. Being different from the displacement approach which employs the kinematic formulation, the approximations based on the stress field satisfy automatically volumetric locking phenomena. Dealing with optimization problems, a second-order cone programming, providing significant advantages of the conic representation for yield criteria, is employed. The examination of various numerical benchmark problems shows an efficient and reliable method for the proposed approach.

关键词： Limit analysis conic programming isogeometric analysis Bezier extraction

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