This paper focusses on the performance of the Metropolis algorithm when employed for solving combinatorial optimization problems. One finds in the literature two notions of success for the Metropolis algorithm in the context of such problems. First, we show that both these notions are equivalent. Next, we provide two characterizations, or in other words, necessary and sufficient conditions, for the success of the algorithm, both characterizations being conditions on the family of Markov chains which the Metropolis algorithm gives rise to when applied to an optimization problem. The first characterization is that the Metropolis algorithm is successful iff in every chain, for every set A of states not containing the optimum, the ratio of the ergodic flow out of A to the capacity of A is high. The second characterization is that in every chain the stationary probability of the optimum is high and that the family of chains mixes rapidly. We illustrate the applicability of our results by giving alternative proofs of certain known results.

This research work intends to examine the resource allocation issues within the hybrid multi‐carrier non‐orthogonal multiple access (MC‐NOMA) systems, which includes the NOMA and orthogonal multiple access (OMA) modes. This is exploited to attain the energy efficiency (EE) and spectral efficiency (SE) tradeoff with the minimum rate requirements of users. The entire degree of freedom that is incorporated within the resource allocation comprised user clustering, power allocation, choice of multiple access (MA) modes, and subcarrier assignment as well. This work mainly focuses on the resource allocation issue having every minimum rate requirement of users for the SE‐EE tradeoff in hybrid MC‐NOMA systems. Moreover, this work aims to make this possible with the incorporation of the optimisation aspect since it is the most significant way of solving multi‐objective problems. Here, the resource allocation along with power allocation seems to be the crucial fact that is modelled or designed as the single‐objective problem. Thereby, the subcarriers, and powers are optimally allocated by means of a new algorithm named Lion algorithm with Probabilistic Mating, which is the modification of the Lion algorithm. Finally, the performance of the proposed work is compared over other state‐of‐the‐art methods in terms of cost analysis.