Most runtime analyses of randomised search heuristics .ocus on the expected number of function valuations to find a unique global optimum. We ask a fundame..tal question: if additional search points are declared optim...
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Most runtime analyses of randomised search heuristics .ocus on the expected number of function valuations to find a unique global optimum. We ask a fundame..tal question: if additional search points are declared optimal, or declared as desirable target points, do these additional optima speed up evolutionary algorithms? More formally, we analyse the expected hitting time of a target set OPUS where S is a set of non-optimal search points and OPI the set of optima and compare it to the expected hitting time of OPT. We show that the answer to our question depends on the number and placement of search points in S. For all black-box algorithms and all fitness functions with polynomial expected optimisation times we show that, if additional optima are placed randomly, even an exponential number of optima has a negligible effect on the expected optimisation time. Considering *** balls around all global optima gives an easier target for some algorithm and functions and can shift the phase transition with respect to offspring population sites in the (1,A) EA on ONE! *AX. However, for the one-dimensional Ising model the time to reach Hamming halls of radius (1/2) around optima does not reduce the asymptotic expected optimisation time in the worst case. Finally, on functions where search trajectories typically join in a single search point, turning one search point into an optimum drastically reduces the expected optimisation time. (c) 2023 Elsevier B.V. All rights reserved.
The discrete logarithm problem over a multiplicative group of integer modulo n is the key ingredient in the ElGamal encryption system. This problem has been proven to be tractable in polynomial time on a quantum compu...
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The discrete logarithm problem over a multiplicative group of integer modulo n is the key ingredient in the ElGamal encryption system. This problem has been proven to be tractable in polynomial time on a quantum computer by Shor's algorithm. This algorithm makes use of basic quantum gates, circuits, measurements and it has been experimented in the circuit-gate framework of quantum computing. On the other hand, the adiabatic framework of quantum computing with the quantum annealers such as the D-Wave machine that simulates the adiabatic process has become very popular, showing its potential for making a practical pathway to achieve quantum speedup. Furthermore, much progress has been reported recently on the tractability of the integer factoring problem, which is the other widely-used encryption method, on quantum annealers. In this context, it is important to explore the tractability of the discrete logarithm problem too using quantum annealers. In this work we present a first step made in this direction, by converting the discrete logarithm problem over a multiplicative group into the problem of minimizing a binary quadratic form, a standard problem format acceptable to a quantum annealer. Our formulation was tested for small-scale problem instances using the quantum-classical hybrid platform PyQUBO and we discuss the computational challenges and propose several potential improvements to the formulation.
A branch-and-bound method is proposed for the maximization of real valued functions with variables assuming only the values 0 and 1. The importance of the problem consists-as has been shown by Hammer and Rudeanu-in th...
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