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Optimization and Control (math.OC)

Wed, 17 May 2023

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1.Solving the problem of batch deletion and insertion members in the Logical Key Hierarchy structure by a DC Programming approach

Authors:Hoai An Le Thi, Thi Tuyet Trinh Nguyen

Abstract: In secure group communications, users of a group share a common group key to prevent eavesdropping and protect the exchange content. A key server distributes the group key as well as performs group rekeying whenever the membership changes dynamically. Instead of rekeying after each join or leave request, we use batch rekeying to alleviate the out-of-sync problem and improve the efficiency. In this paper, we propose an optimization approach to the problem of updating group key in the Logical Key Hierarchy (LKH) structure with batch rekeying. A subtree of new nodes can be appended below a leaf node or is replaced the position of leaving node on the binary key tree. The latter has a lower updating key cost than the former since when a member leaves, all the keys on the path from the root to the deletion node must be updated anyway. We aim to minimize the total rekeying cost, which is the cost of deletion and insertion members while keeping the tree as balanced as possible. The mentioned problem is represented by a unified (deterministic) optimization model whose objective function contains discontinuous step functions with binary variables. Thanks to an exact penalty technique, the problem is equivalently reformulated as a standard DC (Difference of Convex functions) program that can be solved efficiently by DCA (DC algorithm). Numerical experiments have been studied intensively to justify the merit of our proposed approach as well as the corresponding DCA.

2.Algorithms for Boolean Matrix Factorization using Integer Programming

Authors:Christos Kolomvakis, Arnaud Vandaele, Nicolas Gillis

Abstract: Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. As opposed to binary matrix factorization which uses standard arithmetic, BMF uses the Boolean OR and Boolean AND operations to perform matrix products, which leads to lower reconstruction errors. BMF is an NP-hard problem. In this paper, we first propose an alternating optimization (AO) strategy that solves the subproblem in one factor matrix in BMF using an integer program (IP). We also provide two ways to initialize the factors within AO. Then, we show how several solutions of BMF can be combined optimally using another IP. This allows us to come up with a new algorithm: it generates several solutions using AO and then combines them in an optimal way. Experiments show that our algorithms (available on gitlab) outperform the state of the art on medium-scale problems.

3.Computing Optimal Strategies for a Search Game in Discrete Locations

Authors:Jake Clarkson, Kyle Y Lin

Abstract: Consider a two-person zero-sum search game between a hider and a searcher. The hider hides among $n$ discrete locations, and the searcher successively visits individual locations until finding the hider. Known to both players, a search at location $i$ takes $t_i$ time units and detects the hider -- if hidden there -- independently with probability $\alpha_i$, for $i=1,\ldots,n$. The hider aims to maximize the expected time until detection, while the searcher aims to minimize it. We present an algorithm to compute an optimal strategy for each player. We demonstrate the algorithm's efficiency in a numerical study, in which we also study the characteristics of the optimal hiding strategy.

4.Cost-Aware Bound Tightening for Constraint Screening in AC OPF

Authors:Mohamed Awadalla, François Bouffard

Abstract: The objective of electric power system operators is to determine cost-effective operating points by resolving optimization problems that include physical and engineering constraints. As empirical evidence and operator experience indicate, only a small portion of these constraints are found to be binding during operations. Several optimization-based methods have been developed to screen out redundant constraints in operational planning problems like the optimal power flow (OPF) problem. These elimination procedures primarily focus on the feasible region and ignore the role played by the problem's objective function. This letter addresses the constraint screening problem using the bound tightening technique in the context of the OPF problem formulated with a full ac power flow characterization. Due to the non-convexity of the ac OPF, we investigate line constraint screening under different convex relaxations of the problem, and we evaluate how the economics of the objective function impacts screening outcomes.