“Cone-free” primal-dual path-following and potential-reduction polynomial time interior-point methods

We present a framework for designing and analyzing primal-dual interior-point methods for convex optimization. We assume that a self-concordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory and techniques of interior-point methods to the given good formulation of the problem (as is, without a conic reformulation) using the very usual primal central path concept and a less usual version of a dual path concept. We show that many of the advantages of the primal-dual interior-point techniques are available to us in this framework and therefore, they are not intrinsically tied to the conic reformulation and the logarithmic homogeneity of the underlying barrier function.Part of the research was done while the author was a Visiting Professor at the University of Waterloo.Research of this author is supported in part by a PREA from Ontario and by a NSERC Discovery Grant. Tel: (519) 888-4567 ext.5598, Fax: (519) 725-5441Mathematics Subject Classification (2000): 90C51, 90C25, 65Y20,90C28, 49D49     

Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds

Dini derivatives in Riemannian manifold settings are studied in this paper. In addition, a characterization for Lipschitz and convex functions defined on Riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the Dini derivative are given.     

Central Paths in Semidefinite Programming, Generalized Proximal-Point Method and Cauchy Trajectories in Riemannian Manifolds

The relationships among the central path in the context of semidefinite programming, generalized proximal-point method and Cauchy trajectory in a Riemannian manifolds is studied in this paper. First, it is proved that the central path associated to a general function is well defined. The convergence and characterization of its limit point is established for functions satisfying a certain continuity property. Also, the generalized proximal-point method is considered and it is proved that the correspondingly generated sequence is contained in the central path. As a consequence, both converge to the same point. Finally, it is proved that the central path coincides with the Cauchy trajectory in a Riemannian manifold. This work was supported in part by CNPq Grant 302618/2005-8, by PRONEX(CNPq), CAPES-PICDT and FUNAPE/UFG.     

Research Article: On Extending Primal-Dual Interior-Point Method for Linear Optimization to Convex Quadratic Symmetric Cone Optimization

Disjoint frames are interesting frames in Hilbert spaces, which were introduced by Han and Larson in [4 D. Han and D. R. Larson ( 2000 ). Frames, Basis and Group Representations . Memoirs of the American Mathematical Society, No. 679. AMS, Providence, RI.  [Google Scholar]]. In this article, we use disjoint frames to construct frames. In particular, we obtain some conditions for the linear combinations of frames to be frames where the coefficients in the combination may be operators. Our results generalize the corresponding results obtained by Han and Larson. Finally, we provide some examples to illustrate our constructions.     

Penalty and Barrier Methods for Convex Semidefinite Programming

In this paper we present penalty and barrier methods for solving general convex semidefinite programming problems. More precisely, the constraint set is described by a convex operator that takes its values in the cone of negative semidefinite symmetric matrices. This class of methods is an extension of penalty and barrier methods for convex optimization to this setting. We provide implementable stopping rules and prove the convergence of the primal and dual paths obtained by these methods under minimal assumptions. The two parameters approach for penalty methods is also extended. As for usual convex programming, we prove that after a finite number of steps all iterates will be feasible.     

The isoperimetric profile of a smooth Riemannian manifold for small volumes

We define a new class of submanifolds called pseudo-bubbles, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a Riemannian manifold, there is a unique family of concentric pseudo-bubbles which contains all the pseudo-bubbles C ^2,α -close to small spheres. This permit us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension. Work supported by a grant from INDAM “Istituto Nazionale di Alta Matematica Francesco Severi” (Roma).     

A Class of Large-Update and Small-Update Primal-Dual Interior-Point Algorithms for Linear Optimization

In this paper we present a class of polynomial primal-dual interior-point algorithms for linear optimization based on a new class of kernel functions. This class is fairly general and includes the classical logarithmic function, the prototype self-regular function, and non-self-regular kernel functions as special cases. The analysis of the algorithms in the paper follows the same line of arguments as in Bai et al. (SIAM J. Optim. 15:101–128, [2004]), where a variety of non-self-regular kernel functions were considered including the ones with linear and quadratic growth terms. However, the important case when the growth term is between linear and quadratic was not considered. The goal of this paper is to introduce such class of kernel functions and to show that the interior-point methods based on these functions have favorable complexity results. They match the currently best known iteration bounds for the prototype self-regular function with quadratic growth term, the simple non-self-regular function with linear growth term, and the classical logarithmic kernel function. In order to achieve these complexity results, several new arguments had to be used. This research is partially supported by the grant of National Science Foundation of China 10771133 and the Program of Shanghai Pujiang 06PJ14039.     

Invariance and efficiency of convex representations

We consider two notions for the representations of convex cones G-representation and lifted-G-representation. The former represents a convex cone as a slice of another; the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones are invariant under one notion of representation but not the other. In particular, we prove that lifted-G-representation is closed under duality when the representing cone is self-dual. We also prove that strict complementarity of a convex optimization problem in conic form is preserved under G-representations. Then we move to study efficiency measures for representations. We evaluate the representations of homogeneous convex cones based on the “smoothness” of the transformations mapping the central path of the representation to the central path of the represented optimization problem. Research of the first author was supported in part by a grant from the Faculty of Mathematics, University of Waterloo and by a Discovery Grant from NSERC. Research of the second author was supported in part by a Discovery Grant from NSERC and a PREA from Ontario, Canada.     

Geometric Condition Measures and Smoothness Condition Measures for Closed Convex Sets and Linear Regularity of Infinitely Many Closed Convex Sets

In this paper, we study geometric condition measures and smoothness condition measures of closed convex sets, bounded linear regularity, and linear regularity. We show that, under certain conditions, the constant for the linear regularity of infinitely many closed convex sets can be characterized by the geometric condition measure of the intersection or by the smoothness condition measure of the intersection. We study also the bounded linear regularity and present some interesting properties of the general linear regularity problem.The author is grateful to the referees for valuable and constructive suggestions. In particular, she thanks a referee for drawing her attention to Corollary 5.14 of Ref. 3, which inspired her to derive Theorem 4.2 and Corollary 4.2 in the revision of this paper.     

Plurisubharmonic Functions on the Octonionic Plane and <Emphasis Type="Italic">Spin</Emphasis>(9)-Invariant Valuations on Convex Sets

A new class of plurisubharmonic functions on the octonionic plane is introduced. An octonionic version of theorems of A.D. Aleksandrov (Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 13(1):5–24, 1958) and Chern-Levine-Nirenberg (Global Analysis, pp. 119–139, 1969), and Błocki (Proc. Am. Math. Soc. 128(12):3595–3599, 2000) are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of . In particular, a new example of Spin(9)-invariant valuation on ℝ¹⁶ is given. Partially supported by ISF grant 1369/04.     

Characterization of the Nonemptiness and Compactness of Solution Sets in Convex and Nonconvex Vector Optimization

As a consequence of an abstract theorem proved elsewhere, a vector Weierstrass theorem for the existence of a weakly efficient solution without any convexity assumption is established. By using the notion (recently introduced in an earlier paper) of semistrict quasiconvexity for vector functions and assuming additional structure on the space, new existence results encompassing many results appearing in the literature are derived. Also, when the cone defining the preference relation satisfies some mild assumptions (but including the polyhedral and icecream cones), various characterizations for the nonemptiness and compactness of the weakly efficient solution set to convex vector optimization problems are given. Similar results for a class of nonconvex problems on the real line are established as well.Research supported in part by Conicyt-Chile through FONDECYT 104-0610 and FONDAP-Matemáticas Aplicadas II.     

Reduction,Linearization, and Stability of Relative Equilibria for Mechanical Systems on Riemannian Manifolds

Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an explicit characterization is given for the manner in which reduction by an infinitesimal isometry, and linearization along a controlled trajectory “commute.” As part of the development, relationships are derived between the Jacobi equation of geodesic variation and concepts from reduction theory, such as the curvature of the mechanical connection and the effective potential. As an application of our techniques, fiber and base stability of relative equilibria are studied. The paper also serves as a tutorial of Riemannian geometric methods applicable in the intersection of mechanics and control theory. F. Bullo’s research supported in part by grant CMS 0442041 from the USA National Science Foundation. A.D. Lewis’ research supported in part by grant from the Natural Sciences and Engineering Research Council of Canada.     

Globally Convergent Optimization Algorithms on Riemannian Manifolds: Uniform Framework for Unconstrained and Constrained Optimization

This paper proposes several globally convergent geometric optimization algorithms on Riemannian manifolds, which extend some existing geometric optimization techniques. Since any set of smooth constraints in the Euclidean space R ⁿ (corresponding to constrained optimization) and the R ⁿ space itself (corresponding to unconstrained optimization) are both special Riemannian manifolds, and since these algorithms are developed on general Riemannian manifolds, the techniques discussed in this paper provide a uniform framework for constrained and unconstrained optimization problems. Unlike some earlier works, the new algorithms have less restrictions in both convergence results and in practice. For example, global minimization in the one-dimensional search is not required. All the algorithms addressed in this paper are globally convergent. For some special Riemannian manifold other than R ⁿ , the new algorithms are very efficient. Convergence rates are obtained. Applications are discussed. This paper is based on part of the Ph.D Thesis of the author under the supervision of Professor Tits, University of Maryland, College Park, Maryland. The author is in debt to him for invaluable suggestions on earlier versions of this paper. The author is grateful to the Associate Editor and anonymous reviewers, who pointed out a number of papers that have been included in the references; they made also detailed suggestions that lead to significant improvements of the paper. Finally, the author thanks Dr. S.T. Smith for making available his Ph.D Thesis.     

Convex feasibility modeling and projection methods for sparse signal recovery

A computationally-efficient method for recovering sparse signals from a series of noisy observations, known as the problem of compressed sensing (CS), is presented. The theory of CS usually leads to a constrained convex minimization problem. In this work, an alternative outlook is proposed. Instead of solving the CS problem as an optimization problem, it is suggested to transform the optimization problem into a convex feasibility problem (CFP), and solve it using feasibility-seeking sequential and simultaneous subgradient projection methods, which are iterative, fast, robust and convergent schemes for solving CFPs. As opposed to some of the commonly-used CS algorithms, such as Bayesian CS and Gradient Projections for sparse reconstruction, which become inefficient as the problem dimension and sparseness degree increase, the proposed methods exhibit robustness with respect to these parameters. Moreover, it is shown that the CFP-based projection methods are superior to some of the state-of-the-art methods in recovering the signal’s support. Numerical experiments show that the CFP-based projection methods are viable for solving large-scale CS problems with compressible signals.     

Smoothing Technique and its Applications in Semidefinite Optimization

In this paper we extend the smoothing technique (Nesterov in Math Program 103(1): 127–152, 2005; Nesterov in Unconstrained convex mimimization in relative scale, 2003) onto the problems of semidefinite optimization. For that, we develop a simple framework for estimating a Lipschitz constant for the gradient of some symmetric functions of eigenvalues of symmetric matrices. Using this technique, we can justify the Lipschitz constants for some natural approximations of maximal eigenvalue and the spectral radius of symmetric matrices. We analyze the efficiency of the special gradient-type schemes on the problems of minimizing the maximal eigenvalue or the spectral radius of the matrix, which depends linearly on the design variables. We show that in the first case the number of iterations of the method is bounded by \(O({1}/{\epsilon})\), where \(\epsilon\) is the required absolute accuracy of the problem. In the second case, the number of iterations is bounded by \({({4}/{\delta})} \sqrt{(1 + \delta) r\, \ln r }\), where δ is the required relative accuracy and r is the maximal rank of corresponding linear matrix inequality. Thus, the latter method is a fully polynomial approximation scheme.     

Nonlinear rescaling and proximal-like methods in convex optimization

The nonlinear rescaling principle (NRP) consists of transforming the objective function and/or the constraints of a given constrained optimization problem into another problem which is equivalent to the original one in the sense that their optimal set of solutions coincides. A nonlinear transformation parameterized by a positive scalar parameter and based on a smooth sealing function is used to transform the constraints. The methods based on NRP consist of sequential unconstrained minimization of the classical Lagrangian for the equivalent problem, followed by an explicit formula updating the Lagrange multipliers. We first show that the NRP leads naturally to proximal methods with an entropy-like kernel, which is defined by the conjugate of the scaling function, and establish that the two methods are dually equivalent for convex constrained minimization problems. We then study the convergence properties of the nonlinear rescaling algorithm and the corresponding entropy-like proximal methods for convex constrained optimization problems. Special cases of the nonlinear rescaling algorithm are presented. In particular a new class of exponential penalty-modified barrier functions methods is introduced. Partially supported by the National Science Foundation, under Grants DMS-9201297, and DMS-9401871. Partially supported by NASA Grant NAG3-1397 and NSF Grant DMS-9403218.     

Convex Variational Formulation with Smooth Coupling for Multicomponent Signal Decomposition and Recovery

A convex variational formulation is proposed to solve multicomponent signal processing problems in Hilbert spaces. The cost function consists of a separable term, in which each component is modeled through its own potential, and of a coupling term, in which constraints on linear transformations of the components are penalized with smooth functionals. An algorithm with guaranteed weak convergence to a solution to the problem is provided. Various multicomponent signal decomposition and recovery applications are discussed.     

A fast dual proximal gradient algorithm for convex minimization and applications

We consider the convex composite problem of minimizing the sum of a strongly convex function and a general extended valued convex function. We present a dual-based proximal gradient scheme for solving this problem. We show that although the rate of convergence of the dual objective function sequence converges to the optimal value with the rate O(1/k²)$O(1/k^2)$, the rate of convergence of the primal sequence is of the order O(1/k)$O(1/k)$.     

两个凸紧集的差及Clarke广义Jacobi与拟微分的关系

将犚狀空间中两个凸紧集的Ｄｅｍｙａｎｏｖ差推广到犚犿×狀空间．借助于这种差，建立了Ｃｌａｒｋｅ广义Ｊａｃｏｂｉ与拟微分的关系，从而给出了利用拟微分计算Ｃｌａｒｋｅ广义Ｊａｃｏｂｉ的方法．对于两个有限点集凸包给出了它们Ｄｅｍｙａｎｏｖ差的具体表达式．最后讨论了在求解非光滑方程组中的应用．     

Relaxation Methods for Strictly Convex Regularizations of Piecewise Linear Programs

We give an algorithm for minimizing the sum of a strictly convex function and a convex piecewise linear function. It extends several dual coordinate ascent methods for large-scale linearly constrained problems that occur in entropy maximization, quadratic programming, and network flows. In particular, it may solve exact penalty versions of such (possibly inconsistent) problems, and subproblems of bundle methods for nondifferentiable optimization. It is simple, can exploit sparsity, and in certain cases is highly parallelizable. Its global convergence is established in the recent framework of B -functions (generalized Bregman functions). Accepted 29 October 1996