[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

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Convex Analysisの二,三の進展について

In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered includes those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include well-studied cases from many technical fields such as sparse vectors (signal processing, statistics) and low-rank matrices (control, statistics), as well as several others including sums of a few permutation matrices (ranked elections, multiobject tracking), low-rank tensors (computer vision, neuroscience), orthogonal matrices (machine learning), and atomic measures (system identification). The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial structure of the atomic norm ball carries a number of favorable properties that are useful for recovering simple models, and an analysis of the underlying convex geometry provides sharp estimates of the number of generic measurements required for exact and robust recovery of models from partial information. These estimates are based on computing the Gaussian widths of tangent cones to the atomic norm ball. When the atomic set has algebraic structure the resulting optimization problems can be solved or approximated via semidefinite programming. The quality of these approximations affects the number of measurements required for recovery, and this tradeoff is characterized via some examples. Thus this work extends the catalog of simple models (beyond sparse vectors and low-rank matrices) that can be recovered from limited linear information via tractable convex programming.

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The Convex Geometry of Linear Inverse Problems

Convex Analysis: (pms-28)

The role of big data in smart city

In this paper, we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or lift of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual. This generalizes a theorem of Yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts of convex sets can also be characterized similarly. When the cones live in a family, our results lead to the definition of the rank of a convex set with respect to this family. We present results about this rank in the context of cones of positive semidefinite matrices. Our methods provide new tools for understanding cone lifts of convex sets.

Lifts of Convex Sets and Cone Factorizations

Inspired by a question of Lovasz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal called theta bodies of the ideal. These relaxations generalize Lovasz's construction of the theta body of a graph. We establish a relationship between theta bodies and Lasserre's relaxations for real varieties which allows, in many cases, for theta bodies to be expressed as feasible regions of semidefinite programs. Examples from combinatorial optimization are given. Lovasz asked to characterize ideals for which the first theta body equals the closure of the convex hull of its real variety. We answer this question for vanishing ideals of finite point sets via several equivalent characterizations. We also give a geometric description of the first theta body for all ideals.

Theta Bodies for Polynomial Ideals

In this paper we propose an improved mathematical model for locating EV charging stations. We consider the successive activities of the travelers, i.e., a person with two main stops during the day should charge the vehicle in just one of the parking spots, hence avoiding double counting the demand. The model is tested for the city of Coimbra (Portugal), where there is a network of nine stations. We conclude that our solution is better than the one that was implemented in reality, moreover we are able to conclude that demand transference has a rather significant impact on the solutions.

A MIP model for locating slow-charging stations for electric vehicles in urban areas accounting for driver tours

Let $$M \in \mathbb {R}^{p \times q}$$M?Rp×q be a nonnegative matrix. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices $$A_i, B_j$$Ai,Bj of size $$k \times k$$k×k such that $$M_{ij} = {{\mathrm{trace}}}(A_i B_j)$$Mij=trace(AiBj). The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedra and information-theoretic applications. In this paper we develop and survey the main mathematical properties of psd rank, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.

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João Gouveia

Papers

Lifts of Convex Sets and Cone Factorizations

Theta Bodies for Polynomial Ideals

Theta Bodies for Polynomial Ideals

A MIP model for locating slow-charging stations for electric vehicles in urban areas accounting for driver tours

Positive semidefinite rank