The maximum flow algorithm for minimizing energy functions of binary variables has become a standard tool in computer vision. In many cases, unary costs of the energy depend linearly on parameter λ. In this paper we study vision applications for which it is important to solve the maxflow problem for different λ’s. An example is a weighting between data and regularization terms in image segmentation or stereo: it is desirable to vary it both during training (to learn λ from ground truth data) and testing (to select best λ using high-knowledge constraints, e.g. user input). We review algorithmic aspects of this parametric maximum flow problem previously unknown in vision, such as the ability to compute all breakpoints of λ and corresponding optimal configurations in finite time. These results allow, in particular, to minimize the ratio of some geometric functionals, such as flux of a vector field over length (or area). Previously, such functionals were tackled with shortest path techniques applicable only in 2D. We give theoretical improvements for “PDE cuts ” [5]. We present experimental results for image segmentation, 3D reconstruction, and the cosegmentation problem. 1.

Abstract. An exact algorithm is proposed for minimum sum-of-squares nonhierarchical clustering, i.e., for partitioning a given set of points from a Euclidean m-space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong. This problem is expressed as a constrained hyperbolic program in 0-1 variables. The resolution method combines an interior point algorithm, i.e., a weighted analytic center column generation method, with branch-and-bound. The auxiliary problem of determining the entering column (i.e., the oracle) is an unconstrained hyperbolic program in 0-1 variables with a quadratic numerator and linear denominator. It is solved through a sequence of unconstrained quadratic programs in 0-1 variables. To accelerate resolution, variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached. Estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region. Proved minimum sum-of-squares partitions are determined for the first time for several fairly large data sets from the literature, including Fisher’s 150 iris. Key words. classification and discrimination, cluster analysis, interior-point methods, combinatorial optimization

Recently, the first combinatorial strongly polynomial algorithms for submodular function minimization have been devised independently by Iwata, Fleischer, and Fujishige and by Schrijver. In this paper, we improve the running time of Schrijver's algorithm by designing a push-relabel framework for submodular function minimization (SFM). We also extend this algorithm to carry out parametric minimization for a strong map sequence of submodular functions in the same asymptotic running time as a single SFM. Applications include an eicient algorithm for finding a lexicographically optimal base.

The traditional discrete multi-tone equalizer is a cascade of a time domain equalizer (TEQ) as a single finite impulse response filter, a multicarrier demodulator as a fast Fourier transform (FFT), and a frequency domain equalizer (FEQ) as a one-tap filter bank. The TEQ shortens the transmission channel impulse response (CIR) to mitigate inter-symbol interference (ISI). Maximum Bit Rate (MBR) and Minimum ISI (Min-ISI) methods achieve higher data rates at the TEQ output than previously published methods. As an alternative to the traditional equalizer, the per-tone equalizer (PTE) moves the TEQ into the FEQ and customizes a multi-tap FEQ for each tone. In this paper, we propose a time domain TEQ filter bank (TEQFB) and single TEQ that demonstrate better data rates at the FEQ output than MBR, Min-ISI, and least-squares PTE methods with standard CIRs, transmit filters, and receive filters. The contributions of this paper are: (1) a model for the signal-to-noise ratio (SNR) at the FFT output that includes ISI, near-end crosstalk, white Gaussian noise, analog-to-digital converter quantization noise and the digital noise floor; (2) data rate optimal time domain per-tone TEQ filter bank and the upper bound on bit rate performance it achieves; and (3) data rate maximization single TEQ design algorithm. SP EDICS: SP 3-TDSL: Telephone Networks and Digital Subscriber Loops. Milos Milosevic *, B. L. Evans and R. Baldick are with the Dept. of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712-1084, L. F. C. Pessoa is with Motorola, Inc.,7700 West Parmer Lane, MD: TX32-PL30, Austin, TX 78729. E-mail: {milos,bevans,baldick}@ece.utexas.edu and Lucio.Pessoa@motorola.com. B. L. Evans was supported by a gift from the Motorola Semiconductor Products S...

A branch and bound algorithm for computing globally optimal solutions to nonconvex nonlinear programs in continuous variables is presented. The algorithm is directly suitable for a wide class of problems arising in chemical engineering design. It can solve problems defined using algebraic functions and twice differentiable transcendental functions, in which finite upper and lower bounds can be placed on each variable. The algorithm uses rectangular partitions of the variable domain and a new bounding program based on convex/concave envelopes and positive definite combinations of quadratic terms. The algorithm is deterministic and obtains convergence with final regions of finite size. The partitioning strategy uses a sensitivity analysis of the bounding program to predict the best variable to split and the split location. Two versions of the algorithm are considered, the first using a local NLP algorithm (MINOS) and the second using a sequence of lower bounding programs in the search fo...

Abstract. We consider the problem of minimizing a fractional quadratic problem involving the ratio of two indefinite quadratic functions, subject to a two-sided quadratic form constraint. This formulation is motivated by the so-called regularized total least squares (RTLS) problem. A key difficulty with this problem is its nonconvexity, and all current known methods to solve it are guaranteed only to converge to a point satisfying first order necessary optimality conditions. We prove that a global optimal solution to this problem can be found by solving a sequence of very simple convex minimization problems parameterized by a single parameter. As a result, we derive an efficient algorithm that produces an ɛ-global optimal solution in a computational effort of O(n3 log ɛ−1). The algorithm is tested on problems arising from the inverse Laplace transform and image deblurring. Comparison to other well-known RTLS solvers illustrates the attractiveness of our new method. Key words. regularized total least squares, fractional programming, nonconvex quadratic optimization, convex programming

Abstract. Total least squares (TLS) is a method for treating an overdetermined system of linear equations Ax ≈ b, where both the matrix A and the vector b are contaminated by noise. Tikhonov regularization of the TLS (TRTLS) leads to an optimization problem of minimizing the sum of fractional quadratic and quadratic functions. As such, the problem is nonconvex. We show how to reduce the problem to a single variable minimization of a function G over a closed interval. Computing a value and a derivative of G consists of solving a single trust region subproblem. For the special case of regularization with a squared Euclidean norm we show that G is unimodal and provide an alternative algorithm, which requires only one spectral decomposition. A numerical example is given to illustrate the effectiveness of our method.

ABSTRACT: A common model for computing the similarity of two stringsXandYof lengthsm andnrespectively, withmn, is to transformXintoYthrough a sequence of edit operations, called an edit sequence. The edit operations are of three types: insertion, deletion, and substitution. A given cost function assigns a weight to each edit operation. The amortized weight for a given edit sequence is the ratio of its weight to its length, and the minimum of this ratio over all edit sequences is the normalized edit distance. Existing algorithms for normalized edit distance computation with proven complexity bounds requireO(mn2)time in the worst-case. We give provably better algorithms: anO(mnlogn)-time algorithm when the cost function is uniform, i.e, the weights of edit operations depend only on the type but not on the individual symbols involved, and anO(mnlogm)-time algorithm when the weights are rational.

One of the most difficult fractional programs encountered so far is the sumof-ratios problem. Contrary to earlier expectations it is much more removed from convex programming than other multi-ratio problems analyzed before. It really should be viewed in the context of global optimization. It proves to be essentially NP-complete in spite of its special structure under the usual assumptions on numerators and denominators. The paper provides a recent survey of applications, theoretical results and various algorithmic approaches for this challenging problem.

Let A be a nonempty finite subset of the plane representing the geographical coordinates of a set of demand points (towns, . . . ), to be served by a facility, whose location within a given region S is sought. Assuming that the unit cost for a 2 A if the facility is located at x 2 S is proportional to dist(x; a) --- the distance from x to a --- and that demand of point a is given by ! a , minimizing the total transportation cost TC(!;x) amounts to solving the Weber problem. In practice, it may be the case, however, that the demand vector ! is not known, and only an estimator ! can be provided. Moreover the errors in such estimation process may be non-negligible. We propose a new model for this situation: select a threshold value B ? 0 representing the highest admissible transportation cost. Define the robustness ae of a location x as the minimum increase in demand needed to become inadmissible, i.e. ae(x) = min fk! \Gamma !k : TC(! ; x) ? B; ! 0g and find the x maximizin...