Abstract. In this paper we present optimization algorithms for image restoration based on the total variation (TV) minimization framework of L. Rudin, S. Osher and E. Fatemi (ROF). Our approach formulates TV minimization as a second-order cone program which is then solved by interior-point algorithms that are efficient both in practice (using nested dissection and domain decomposition) and in theory (i.e., they obtain solutions in polynomial time). In addition to the original ROF minimization model, we show how to apply our approach to other TV models including ones that are not solvable by PDE based methods. Numerical results on a varied set of images are presented to illustrate the effectiveness of our approach.

In this paper, we present the logarithmic total variation (LTV) model for face recognition under varying illumination, including natural lighting condition, where we can hardly know the strength, the directions, and the number of light sources. The proposed LTV model has the capability to factorize a single face image and obtain the illumination invariant facial structure, which is then used for face recognition. The merit of this model is that neither does it require any lighting assumption nor does it need any training process. Besides, there is only one parameter which could be easily set. The LTV model is able to reach very high recognition rates on both Yale and CMU PIE face databases as well as on a face database containing 765 subjects under outdoor lighting conditions. Keywords: I.5.4.d Face and gesture recognition; I.5.4.m Signal processing; I.4 Image Processing and Computer Vision; I.5.2.c Pattern analysis;

this paper can be downloaded from http://www.compgeom.com/meb/. P. Kumar and J. Mitchell are partially supported by a grant from the National Science Foundation (CCR0098172) . J. Mitchell is also partially supported by grants from the Honda Fundamental Research Labs, Metron Aviation, NASAAmes Research (NAG2-1325), and the US-Israel Binational Science Foundation. E. A. Yldrm is partially supported by an NSF CAREER award (DMI-0237415)

In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 0-1 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 0-1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank-1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.

High Dimension Low Sample Size statistical analysis is becoming increasingly important in a wide range of applied contexts. In such situations, it is seen that the popular Support Vector Machine suffers from “data piling ” at the margin, which can diminish generalizability. This leads naturally to the development of Distance Weighted Discrimination, which is based on Second Order Cone Programming, a modern computationally intensive optimization method.

Regularized kernel discriminant analysis (RKDA) performs linear discriminant analysis in the feature space via the kernel trick. Its performance depends on the selection of kernels. In this paper, we consider the problem of multiple kernel learning (MKL) for RKDA, in which the optimal kernel matrix is obtained as a linear combination of pre-specified kernel matrices. We show that the kernel learning problem in RKDA can be formulated as convex programs. First, we show that this problem can be formulated as a semidefinite program (SDP). Based on the equivalence relationship between RKDA and least square problems in the binary-class case, we propose a convex quadratically constrained quadratic programming (QCQP) formulation for kernel learning in RKDA. A semi-infinite linear programming (SILP) formulation is derived to further improve the efficiency. We extend these formulations to the multi-class case based on a key result established in this paper. That is, the multi-class RKDA kernel learning problem can be decomposed into a set of binary-class kernel learning problems which are constrained to share a common kernel. Based on this decomposition property, SDP formulations are proposed for the multi-class case. Furthermore, it leads naturally to QCQP and SILP formulations. As the performance of RKDA depends on the regularization parameter, we show that this parameter can also be optimized in a joint framework with the kernel. Extensive experiments have been conducted and analyzed, and connections to other algorithms are discussed.

The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact proximal point algorithms in the primal, dual and primal-dual forms for solving the nuclear norm minimization with linear equality and second order cone constraints. We design efficient implementations of these algorithms and present comprehensive convergence results. In particular, we investigate the performance of our proposed algorithms in which the inner sub-problems are approximately solved by the gradient projection method or the accelerated proximal gradient method. Our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed state-of-the-art algorithms. Interestingly, our proposed algorithms are connected with other algorithms that have been studied in the literature. Key words. Nuclear norm minimization, proximal point method, rank minimization, gradient projection method, accelerated proximal gradient method.

In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. Unlike [4], our focus in this paper is to identify uncertainty structures that allow the corresponding robust quadratically constrained programs to be reformulated as second-order cone programs. We propose three classes of uncertainty sets that satisfy this criterion and present examples where these classes of uncertainty sets are natural. 1 Problem formulation A generic quadratically constrained program (QCP) is defined as follows.

The active contour/snake model [9, 2, 10] is one of the most wellknown segmentation variational models in image processing. However this model suffers from the existence of local minima which makes the initial guess critical for getting satisfactory results. In this paper, we propose to solve this problem by finding global minimizers of the active contour model following the original work of Chan, Esedo¯glu and Nikolova [4]. Our approach uses the weighted total variation norm to link the standard active contour segmentation model with the denoising model of Rudin-Osher-Fatemi [15] and the Chan-Vese active contour segmentation models [5, 18] based on the Mumford-Shah functional [12]. 1

Parametric yield loss due to variability can be effectively reduced by both design-time optimization strategies and by adjusting circuit parameters to the realizations of variable parameters. The two levels of tuning operate within a single variability budget, and because their effectiveness depends on the magnitude and the spatial structure of variability their joint co-optimization is required. In this paper we develop a formal optimization algorithm for such co-optimization and link it to the control and measurement overhead via the formal notions of measurement and control complexity. We describe an optimization strategy that unifies design-time gate-level sizing and post-silicon adaptation using adaptive body bias at the chip level. The statistical formulation utilizes adjustable robust linear programming to derive the optimal policy for assigning body bias once the uncertain variables, such as gate length and threshold voltage, are known. Computational tractability is achieved by restricting optimal body bias selection policy to be an affine function of uncertain variables. We demonstrate good run-time and show that 5-35 % savings in leakage power across the benchmark circuits are possible. Dependence of results on measurement and control complexity is studied and points of diminishing returns for both metrics are identified.