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This paper introduces a computational method for generating metric Travelling Salesman Problem (TSP) instances having a large integrality gap. The method is based on the solution of an integer programming problem, called IH-OPT, that takes as input a fractional solution of the Subtour Elimination Problem (SEP) on a TSP instance and computes a TSP instance having an integrality gap larger than or equal to the integrality gap of the first instance. The decision variables of IH-OPT are the entries of the TSP cost matrix, and the constraints are defined by the intersection of the metric cone with an exponential number of inequalities, one for each possible TSP tour. |
In this manuscript, a novel general class of contractions, called Jaggi–Suzuki-type hybrid (G-α-ϕ)-contraction, is introduced and some fixed point theorems that cannot be deduced from their akin in metric spaces are proved. The dominance of this family of contractions is that its contractive inequality can be specialized in various manners, depending on multiple parameters. Nontrivial comparative examples are constructed to validate the assumptions of our obtained theorems. Consequently, a number of corollaries that reduce our result to some prominent results in the literature are highlighted and analyzed |
Extending the main result of Lorscheid and Weist (2015), in the first part of this paper we show that every quiver Grassmannian of an indecomposable representation of a quiver of type D~n has a decomposition into affine spaces. In the case of real root representations of small defect, the non-empty cells are in one-to-one correspondence to certain, so called non-contradictory, subsets of the vertex set of a fixed tree-shaped coefficient quiver. In the second part, we use this characterization to determine the generating functions of the Euler characteristics of the quiver Grassmannians (resp. F-polynomials). |
Unit square visibility graphs (USV) are described by axis-parallel visibility between unit squares placed in the plane. If the squares are required to be placed on integer grid coordinates, then USV become unit square grid visibility graphs (USGV), an alternative characterisation of the well-known rectilinear graphs. We extend known combinatorial results for USGV and we show that, in the weak case (i.e., visibilities do not necessarily translate into edges of the represented combinatorial graph), the area minimisation variant of their recognition problem is NP
-hard. |
We present an algorithm aimed to recognize if a given tensor is a non-identifiable rank-3 tensor. |
To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective’s gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. |
In this paper we study removable singularities for solutions of the fractional heat equation in the spacial-time space. We introduce associated capacities and we study some of its metric and geometric properties. |
The concept of subordination, originally introduced in the probability and stochastic processes theories, has also appeared in analysis of evolution equations. So it is not surprising that we meet it in physics of complex systems, in particular when study equations describing diffusion and dielectric relaxation phenomena. Grace to intuitively understood decomposition of complex processes into their simpler and better known components, called parent and leading processes, subordination formalism enables us to attribute physical interpretation to integral decompositions representing plethora of solutions to anomalous diffusion and relaxation problems. |
We develop an exact cutting plane solution algorithm for a special class of bilevel programming models utilized for optimal price-bidding of energy producers in day-ahead electricity markets. The proposed methodology utilizes a suitable reformulation in which a key prerequisite for global optimality, termed bilevel feasibility, is relaxed. Solving the problem to global optimality involves finding the price-offers of the strategic producer (upper-level decision variables) which maximize his self-profit upon clearing of the market and identification of the optimal energy quantity distribution (lower-level decision variables). |
In this paper we address the question if, for points P,Q∈P2, I(P)m⋆I(Q)n=I(P⋆Q)m+n−1 and we obtain different results according to the number of zero coordinates in P and Q. Successively, we use our results to define the so called Hadamard fat grids, which are the result of the Hadamard product of two sets of collinear points with given multiplicities. The most important invariants of Hadamard fat grids, as minimal resolution, Waldschmidt constant and resurgence, are then computed. |
