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### The Analytic-Center Cutting-Plane Method for Variational Inequalities: A Quadratic-Cut Approach

Michel Denault, Jean-Louis Goffin
2005 INFORMS journal on computing
W e introduce a cutting-plane, analytic-center algorithm for strongly monotone variational inequalities (VIs). The approach extends that of Goffin et al. (1997) and Denault and Goffin (1999) .  ...  Our approach uses quadratic, ellipsoidal cuts, based on the symmetrized Jacobian of the VI.  ...  to by the acronym ACCPM, for analytic center cutting plane method.  ...

### Solving variational inequalities with a quadratic cut method: a primal-dual, Jacobian-free approach

Michel Denault, J.-L. Goffin
2004 Computers & Operations Research
We extend in two directions the Analytic Center, Cutting Plane Method for Variational Inequalities with quadratic cuts, ACCPM-VI(quadratic cuts), introduced by Denault and Go n in 1998.  ...  First, we deÿne a primal-dual method to ÿnd the analytic center at each iteration.  ...  Review of cutting plane methods for variational inequalities The ACCPM-VI algorithms (there are many versions) all use, in one way or another, the Analytic Center Cutting Plane Method, pioneered by Go  ...

### On a Primal-Dual Analytic Center Cutting Plane Method for Variational Inequalities [chapter]

M. Denault, J. L. Goffin
1999 Computational Optimization
We present an algorithm for variational inequalities V I(F, Y ) that uses a primal-dual version of the Analytic Center Cutting Plane Method.  ...  Each computation of a new analytic center requires at most four Newton iterations, in theory, and in practice one or sometimes two.  ...  The notation O * indicates that lower order terms are ignored 4. A formal proof of this would require a very significant and lengthy rewriting of the proofs of [12] and [13] .  ...

### Homogeneous Analytic Center Cutting Plane Methods for Convex Problems and Variational Inequalities

Yu. Nesterov, J. Ph. Vial
1999 SIAM Journal on Optimization
In this paper we consider a new analytic center cutting plane method in a projective space.  ...  We prove the e ciency estimates for the general scheme and show that these results can be used in the analysis of a feasibility problem, the variational inequality problem and the problem of constrained  ...  B ueler worked out the computations on the small example of Section 4. We thank him for letting us make use of his results. We are also grateful for his comments on the paper.  ...

### Solving variational inequalities defined on a domain with infinitely many linear constraints

Shu-Cherng Fang, Soonyi Wu, Ş. İlker Birbil
2007 Computational optimization and applications
A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given.  ...  We study a variational inequality problem whose domain is defined by infinitely many linear inequalities.  ...  Recently, the authors in [8, 24] presented an analytic center based cutting plane method for solving a general semi-infinite variational inequality problem.  ...

### A Lagrangian dual method for solving variational inequalities

Stefan M. Stefanov
2002 Mathematical Inequalities & Applications
An analytic center quadratic cut method for strongly monotone variational inequality problems is suggested in Lüthi and Büeler 12].  ...  Mathematics subject classification Nesterov and Vial ( 15]) introduced a homogeneous analytic center cutting plane method (HACCPM) which solves monotone VIPs in a conic setting and pseudopolynomial-time  ...  ;γ k x k =µ k; 1 i 1 ) i = 1 : : : m + 2n: A LAGRANGIAN DUAL METHOD FOR SOLVING VARIATIONAL INEQUALITIES 599 For example, the differential version of KKT theorem for problem min f (x) subject to  ...

### New Method to Enforce Inequality Constraints of Entry Trajectory

Li Huifeng, Zhang Ran, Li Zhaoying, Zhang Rui
2012 Journal of Guidance Control and Dynamics
the need to solve a complex optimiz- ation problem in real time and 2) successfully flight testing the approach through an experimental application to a cutting-edge quadrotor helicopter UAV.  ...  ., “A Direct Method for Approach and Landing Trajectory Reshaping with Failure Effect Estimation,” AIAA Paper 2004-4772, Aug. 2004. [12] Leavitt, J. A., and Mease, K.  ...

### Shallow, deep and very deep cuts in the analytic center cutting plane method

Jean-Louis Goffin, Jean-Philippe Vial
1999 Mathematical programming
The analytic center cutting plane (ACCPM) methods aims to solve nondi erentiable convex problems.  ...  The primal framework leads to a simple analysis of the potential variation, which shows that the inequality needed for convergence of the algorithm is in fact attained at the rst iterate of the feasibility  ...  The analytic center cutting plane method chooses as query point an approximate analytic center of F D .  ...

### Finsler interpolation inequalities

Shin-ichi Ohta
2009 Calculus of Variations and Partial Differential Equations
comparison, optimal transport, curvature-dimension condition, concentration of measure. ‡ Partly supported by the JSPS fellowship for research abroad.  ...  Among applications, we establish the equivalence between Sturm, Lott and Villani's curvature-dimension condition and a certain lower Ricci curvature bound.  ...  If r(v) < ∞, then exp x (r(v)v) is called a cut point of x, and the cut locus Cut(x) of x is defined as the set of all cut points of x.  ...

### Page 3362 of Mathematical Reviews Vol. , Issue 98E [page]

1998 Mathematical Reviews
Also, his analysis shows that a new approximate analytic center can be obtained in O(1) Newton iterations, just like the cutting plane method which uses a single cut at each step.  ...  The author considers the analytic center cutting plane (or column generation) algorithm for solving general convex problems defined by a separation oracle.  ...

### On the inequalities of Babuška-Aziz, Friedrichs and Horgan-Payne [article]

Martin Costabel
2013 arXiv   pre-print
For the Horgan-Payne inequality, which is an upper bound of the Friedrichs constant for plane star-shaped domains in terms of a geometric quantity known as the Horgan-Payne angle, we show that it is true  ...  The equivalence between the inequalities of Babuška-Aziz and Friedrichs for sufficiently smooth bounded domains in the plane has been shown by Horgan and Payne 30 years ago.  ...  We want to thank our colleagues Michel Crouzeix, Christine Bernardi, Vivette Girault and Fédéric Hecht for stimulating discussions.  ...

### Page 6962 of Mathematical Reviews Vol. , Issue 2002I [page]

2002 Mathematical Reviews
Summary: “In this paper, an analytic center based cutting plane method is proposed for solving linear semi-infinite programming problems.  ...  Jongen (D-AACH-M2; Aachen) 2002i:90096 90C34 90C53 Wu, Soon-Yi (RC-TAIN-AM; Tainan); Fang, Shu-Cherng (1-NCS-OI; Raleigh, NC); Lin, Chih-Jen (RC-NTAI-IFE; Taipei) Analytic center based cutting plane method  ...

### Page 4499 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews
The authors consider an analytic center cutting plane method for a variational inequality problem of the following form: Find y* € Y and f* € F(y*) such that (f*)'(y — y*) >0 Vy € Y, where ¥ ={y€R”| A'y  ...  -L. (3-MTRLC-G; Montreal, QC) On a primal-dual analytic center cutting plane method for variational inequalities. (English summary) Computational optimization—a tribute to Olvi Mangasarian, Part I.  ...

### Convex nondifferentiable optimization: A survey focused on the analytic center cutting plane method

Jean-Louis Goffin, Jean-Philippe Vial
2002 Optimization Methods and Software
We present a survey of nondi erentiable optimization problems and methods with special focus on the analytic center cutting plane method.  ...  Furthermore, interior point methods o er highly e cient and robust methods to compute these centers. The analytic center cutting plane method (ACCPM) has been built on that principle.  ...  The analytic center cutting plane. From now on, the paper is devoted to an in-depth analysis of the analytic center cutting plane method. The convergence of the method was rst studied in 3].  ...

### A tutorial on linear and bilinear matrix inequalities

Jeremy G. VanAntwerp, Richard D. Braatz
2000 Journal of Process Control
This is a tutorial on the mathematical theory and process control applications of linear matrix inequalities (LMIs) and bilinear matrix inequalities (BMIs).  ...  Many convex inequalities common in process control applications are shown to be LMIs. Proofs are included to familiarize the reader with the mathematics of LMIs and BMIs.  ...  [22] gives analytical expressions for this cutting plane for each of the LMI problems.  ...
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