On the Linear Extension Complexity of Regular n-gons

In this paper, we propose new lower and upper bounds on the linear extension complexity of regular $n$-gons. Our bounds are based on the equivalence between the computation of (i) an extended formulation of size $r$ of a polytope $P$, and (ii) a rank-$r$ nonnegative factorization of a slack matrix of the polytope $P$. The lower bound is based on an improved bound for the rectangle covering number (also known as the boolean rank) of the slack matrix of the $n$-gons. The upper bound is a slight improvement of the result of Fiorini, Rothvoss and Tiwary [Extended Formulations for Polygons, Discrete Comput. Geom. 48(3), pp. 658-668, 2012]. The difference with their result is twofold: (i) our proof uses a purely algebraic argument while Fiorini et al. used a geometric argument, and (ii) we improve the base case allowing us to reduce their upper bound $2 \left\lceil \log_2(n) \right\rceil$ by one when $2^{k-1} < n \leq 2^{k-1}+2^{k-2}$ for some integer $k$. We conjecture that this new upper bound is tight, which is suggested by numerical experiments for small $n$. Moreover, this improved upper bound allows us to close the gap with the best known lower bound for certain regular $n$-gons (namely, $9 \leq n \leq 13$ and $21 \leq n \leq 24$) hence allowing for the first time to determine their extension complexity.Comment: 20 pages, 3 figures. New contribution: improved lower bound for the boolean rank of the slack matrices of n-gon

Algorithms for Positive Semidefinite Factorization

This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an $m$-by-$n$ nonnegative matrix $X$ and an integer $k$, the PSD factorization problem consists in finding, if possible, symmetric $k$-by-$k$ positive semidefinite matrices $\{A^1,...,A^m\}$ and $\{B^1,...,B^n\}$ such that $X_{i,j}=\text{trace}(A^iB^j)$ for $i=1,...,m$, and $j=1,...n$. PSD factorization is NP-hard. In this work, we introduce several local optimization schemes to tackle this problem: a fast projected gradient method and two algorithms based on the coordinate descent framework. The main application of PSD factorization is the computation of semidefinite extensions, that is, the representations of polyhedrons as projections of spectrahedra, for which the matrix to be factorized is the slack matrix of the polyhedron. We compare the performance of our algorithms on this class of problems. In particular, we compute the PSD extensions of size $k=1+ \lceil \log_2(n) \rceil$ for the regular $n$-gons when $n=5$, $8$ and $10$. We also show how to generalize our algorithms to compute the square root rank (which is the size of the factors in a PSD factorization where all factor matrices $A^i$ and $B^j$ have rank one) and completely PSD factorizations (which is the special case where the input matrix is symmetric and equality $A^i=B^i$ is required for all $i$).Comment: 21 pages, 3 figures, 3 table

Heuristics for Exact Nonnegative Matrix Factorization

The exact nonnegative matrix factorization (exact NMF) problem is the following: given an $m$-by-$n$ nonnegative matrix $X$ and a factorization rank $r$, find, if possible, an $m$-by-$r$ nonnegative matrix $W$ and an $r$-by-$n$ nonnegative matrix $H$ such that $X = WH$. In this paper, we propose two heuristics for exact NMF, one inspired from simulated annealing and the other from the greedy randomized adaptive search procedure. We show that these two heuristics are able to compute exact nonnegative factorizations for several classes of nonnegative matrices (namely, linear Euclidean distance matrices, slack matrices, unique-disjointness matrices, and randomly generated matrices) and as such demonstrate their superiority over standard multi-start strategies. We also consider a hybridization between these two heuristics that allows us to combine the advantages of both methods. Finally, we discuss the use of these heuristics to gain insight on the behavior of the nonnegative rank, i.e., the minimum factorization rank such that an exact NMF exists. In particular, we disprove a conjecture on the nonnegative rank of a Kronecker product, propose a new upper bound on the extension complexity of generic $n$-gons and conjecture the exact value of (i) the extension complexity of regular $n$-gons and (ii) the nonnegative rank of a submatrix of the slack matrix of the correlation polytope.Comment: 32 pages, 2 figures, 16 table

A Homotopy-based Algorithm for Sparse Multiple Right-hand Sides Nonnegative Least Squares

Nonnegative least squares (NNLS) problems arise in models that rely on additive linear combinations. In particular, they are at the core of nonnegative matrix factorization (NMF) algorithms. The nonnegativity constraint is known to naturally favor sparsity, that is, solutions with few non-zero entries. However, it is often useful to further enhance this sparsity, as it improves the interpretability of the results and helps reducing noise. While the $\ell_0$-"norm", equal to the number of non-zeros entries in a vector, is a natural sparsity measure, its combinatorial nature makes it difficult to use in practical optimization schemes. Most existing approaches thus rely either on its convex surrogate, the $\ell_1$-norm, or on heuristics such as greedy algorithms. In the case of multiple right-hand sides NNLS (MNNLS), which are used within NMF algorithms, sparsity is often enforced column- or row-wise, and the fact that the solution is a matrix is not exploited. In this paper, we first introduce a novel formulation for sparse MNNLS, with a matrix-wise $\ell_0$ sparsity constraint. Then, we present a two-step algorithm to tackle this problem. The first step uses a homotopy algorithm to produce the whole regularization path for all the $\ell_1$-penalized NNLS problems arising in MNNLS, that is, to produce a set of solutions representing different tradeoffs between reconstruction error and sparsity. The second step selects solutions among these paths in order to build a sparsity-constrained matrix that minimizes the reconstruction error. We illustrate the advantages of our proposed algorithm for the unmixing of facial and hyperspectral images.Comment: 20 pages + 7 pages supplementary materia

Yard traffic and congestion in container terminals

In container terminals, the yard is the zone where containers are stored: it serves as a buffer for loading, unloading and transshipping containers. Yard is usually the bottleneck of the terminal, as traffic, congestion and capacity issues mainly originate from there. A berth&yard allocation plan assigns ships to berths and containers to yard blocks: a typical objective is to minimize the total distance traveled by the carriers to transfer containers from the quayside to the yard. Although congestion issues are usually disregarded, operations are often slowed down because of overloaded areas in the yard. In particular, we are interested in evaluating and reducing traffic congestion induced by the berth&yard allocation plan at the operational level. In this work, we firstly present quantitative indicators we devised to measure congestion in the terminal network and to capture its effects on terminals efficiency. These measures are then combined to provide new objective functions, other than traveled distance, which are used in the design of berth&yard templates; a local search algorithm which minimizes congestion is presented. Computational results are analyzed and discussed

Sunlight refraction in the mesosphere of Venus during the transit on June 8th, 2004

Many observers in the past gave detailed descriptions of the telescopic aspect of Venus during its extremely rare transits across the Solar disk. In particular, at the ingress and egress, the portion of the planet's disk outside the Solar photosphere has been repeatedly perceived as outlined by a thin, bright arc ("aureole"). Those historical visual observations allowed inferring the existence of Venus' atmosphere, the bright arc being correctly ascribed to the refraction of light by the outer layers of a dense atmosphere. On June 8th, 2004, fast photometry based on electronic imaging devices allowed the first quantitative analysis of the phenomenon. Several observers used a variety of acquisition systems to image the event -- ranging from amateur-sized to professional telescopes and cameras -- thus collecting for the first time a large amount of quantitative information on this atmospheric phenomenon. In this paper, after reviewing some elements brought by the historical records, we give a detailed report of the ground based observations of the 2004 transit. Besides confirming the historical descriptions, we perform the first photometric analysis of the aureole using various acquisition systems. The spatially resolved data provide measurements of the aureole flux as a function of the planetocentric latitude along the limb. A new differential refraction model of solar disk through the upper atmosphere allows us to relate the variable photometry to the latitudinal dependency of scale-height with temperature in the South polar region, as well as the latitudinal variation of the cloud-top layer altitude. We compare our measurements to recent analysis of the Venus Express VIRTIS-M, VMC and SPICAV/SOIR thermal field and aerosol distribution. Our results can be used a starting point for new, more optimized experiments during the 2012 transit event.Comment: Icarus, in pres