Quantum computing is an emerging field that have given a significant impact on optimization. Among the diverse quantum algorithm, the quantum gradient descent has become a prominent technique for solving unconstrained optimization problems. In this paper, we propose a quantum-spectral Polak-Ribi{\'e}re-Polyak (PRP) conjugate gradient approach. The technique is considered as a generalization of the spectral PRP method which employs a $q$-gradient that approximate the classical gradient with quadratically better dependence on the quantum variable $q$. Additionally, the proposed method reduces to the classical variant as quantum variable $q$ gets closer to $1$. The quantum search direction always satisfies the sufficient descent condition and does not depend on any line search. This approach is globally convergent with the standard Wolfe line search without any convexity assumption. Numerical experiments are conducted and compared with the existing approach to demonstrate the improvement of the proposed strategy.

The quadratic unconstrained binary optimization (QUBO) is a classic NP-hard problem with an enormous number of applications. Local search strategy (LSS) is one of the most fundamental algorithmic concepts that has been successfully applied to a wide range of hard combinatorial optimization problems. One LSS that has gained the attention of researchers is the r-flip (also known as r-Opt) strategy. Given a binary solution with n variables, the r-flip strategy ‘flips’ r binary variables to get a new solution if the changes improve the objective function. The main purpose of this paper is to develop several results for implementation of r-flip moves in QUBO, including a necessary and sufficient condition that when a 1-flip search reaches local optimality, the number of candidates for implementation of the r-flip moves can be reduced significantly. The results of the substantial computational experiments are reported to compare an r-flip strategy embedded algorithm and a multiple start tabu search algorithm on a set of benchmark instances and three very-large-scale QUBO instances. The r-flip strategy implemented within the algorithm makes the algorithm very efficient, very high-quality solutions within a short CPU time.

The influence of neutrosophy in the previous period is constantly growing in many areas of science and technology. Moreover, various applications of the neutrosophic approach have become more common in recent years. Our goal in this research is to utilize the neutrosophy to improve the performance of the Dai-Liao conjugate gradient (CG) method. Specifically, in this research, we propose and investigate a new neutrosophic logic system to calculate the key parameter t involved in the Dai–Liao CG iterations. Theoretical analysis and numerical experience indicate that the efficiency and robustness of the new rule for determining t. Combining the neutrosophy and the Dai-Liao conjugate gradient method, we propose and explore a new Dai-Liao CG iterations for solving large-scale unconstrained optimization models. The global convergence is established under common assumptions and the backtracking line search. Finally, by conducting numerical experiments, computational evidence demonstrates that the new fuzzy neutrosophic Dai-Liao conjugate gradient method is computationally effective and robust.

Quantum optimization is a significant area of quantum computing with anticipated near-term quantum advantages. Current quantum optimization algorithms, most of which are hybrid variational Hamiltonian-based algorithms, struggle on present quantum devices due to noise and decoherence. Existing techniques attempt to mitigate these issues through employing different Hamiltonian encodings or Hamiltonian clause pruning but often rely on optimistic assumptions rather than a deep analysis of the problem structure. We demonstrate how to formulate the problem Hamiltonian for the Quantum Approximate Optimization Algorithm to satisfy all the requirements that correctly describe the considered tactical aircraft deconfliction problem, achieving higher probabilities of measuring solutions compared to previous works. Our results indicate that constructing Hamiltonians from an unconventional, quantum-specific perspective with a high degree of entanglement results in linear number of entangleming gates instead of exponential and superior performance compared to standard formulations. Specifically, we achieve higher probabilities of measuring feasible solutions in 9 out of 9 instances compared to standard Hamiltonian formulations and surpass quadratic programming formulations known from quantum annealers in 7 out of 9 instances. These findings suggest that there is substantial potential for further research in quantum Hamiltonian design and that gate-based approaches may offer superior optimization performance over quantum annealers in the future.