arXiv subjects

Beniamin Costandin

Publications and source records attributed to Beniamin Costandin.

5 recordsLinked to original sources

arXiv2024

Tight Bounds for the Maximum Distance Over a Polytope to a Given Point

In this paper we study the problem of maximizing the distance to a given point $C_0$ over a polytope $\mathcal{P}$. Assuming that the polytope is circumscribed by a known ball we construct an intersection of balls which preserves the vertices of the polytope on the boundary of this ball, and show that the intersection of balls approximates the polytope arbitrarily well. Then, we use some known results regarding the maximization of distances to a given point over an intersection of balls to create a new polytope which preserves the maximizers to the original problem. Next, a new intersection of balls is obtained in a similar fashion, and as such, after a finite number of iterations, we conjecture, we end up with an intersection of balls over which we can maximize the distance to the given point. The obtained distance is shown to be a non trivial upper bound to the original distance. Tests are made with maximizing the distance to a random point over the unit hypercube up to dimension $n = 100$. Several detailed 2-d examples are also shown.

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arXiv2024

On Efficient Approximation of the Maximum Distance to A Point Over an Intersection of Balls

In this paper we study the NP-Hard problem of maximizing the distance over an intersection of balls to a given point. We expand the results found in \cite{funcos1}, where the authors characterize the farthest in an intersection of balls $\mathcal{Q}$ to the given point $C_0$ by constructing some intersection of halfspaces. In this paper, by slightly modifying the technique found in literature, we characterize the farthest in an intersection of balls $\mathcal{Q}$ with another intersection of balls $\mathcal{Q}_1$. As such, going backwards, we are naturally able to find the given intersection of balls $\mathcal{Q}$ as the max indicator intersection of balls of another one $\mathcal{Q}_{-1}$. By repeating the process, we find a sequence of intersection of balls $(\mathcal{Q}_{i})_{i \in \mathbb{Z}}$, which has $\mathcal{Q}$ as an element, namely $\mathcal{Q}_{0}$ and show that $\mathcal{Q}_{-\infty} = \mathcal{B}(C_0,R_0)$ where $R_0$ is the maximum distance from $C_0$ to a point in $\mathcal{Q}$. As a final application of the proposed theory we give a polynomial algorithm for computing the maximum distance under an oracle which returns the volume of an intersection of balls, showing that the later is NP-Hard. Finally, we present a randomized method %of polynomial complexity which allows an approximation of the maximum distance.

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arXiv2023

A Quasi-Polynomial Algorithm for Subset-Sum Problems with At Most One Solution

In this paper we study the problem of maximizing the distance to a given point over an intersection of balls. It was already known that this problem can be solved in polynomial time and space if the given point is not in the convex hull of the balls centers. The cases where the given point is in the convex hull of the balls centers include all NP-complete problems as we show. Some novel results are given in this area. A novel projection algorithm is developed then applied in the context of the Subset Sum Problem (SSP). Under the assumption that the SSP has at most one solution, we provide a quasi-polynomial algorithm, which decreases the radius of an initial ball containing the solution to the SSP. We perform some numerical tests which show the effectiveness of the proposed algorithm.

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arXiv2020

Maximizing The Distance To A "Far Enough" Point Over The Intersection Of Hyper-Disks

We present a novel feasibility criteria for the finite intersection of convex sets given by inequalities. This criteria allows us to easily assert the feasibility by analyzing the unconstrained minimum of a speci?fic convex function, that we form with the given sets. Next an algorithm is presented which extends the idea to a particular non-convex case: assert the inclusion of the fi?nite intersection of a set of hyper-disks with equal radii in another hyper-disk with a different radius.

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arXiv2020

Unconstraint minimization of continuous convex functions. Application to LP

Our contribution in this paper is two folded. We consider first the case of linear programming with real coefficients and give a method which allows the computation of a new upper bound on the distance from the origin to a feasible point. Next we present an application of the ellipsoid method to form a novel algorithm which allows one to search for the continuous convex functions minimum without the prior knowledge on the search radius. If stopped early, the proposed algorithm can give proofs that the optimal value has not been reached yet, hence the user can opt for more iterations. However, if prior guarantees exist on the existence of an optimum point in a given ball, the algorithm is guaranteed to find it. For such a case we prove a polynomial upper bound on the number of flops required to obtain the solution. The presented algorithm is then applied to linear programming. We further develop our algorithm and provide a method to answer the feasibility question of linear programs with real coefficients in a number of flops bounded above by a polynomial in the number of variables and the number of constraints. However, in case of feasibility, our method does not generate an actual feasible point. For obtaining such a point, we have to make some assumptions on the subgradients of a certain function.

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