Fast Solving Complete 2000-Node Optimization Using Stochastic-Computing Simulated Annealing

Recently, the large-scale optimization problems have become a common research topic in the field of evolutionary computation. It is hard to find optimal solutions when solving large-scale multi-objective optimization problems (LSMOPs), due to the ineffectiveness of existing operators. In the other word, the search ability of most existing MOEAs on solving LSMOPs is still weak. To address this issue, an efficient competitive swarm optimizer with a strong exploration ability, denoted as E-CSO, is presented in this paper, which designs a novel three-particle-based particle updating strategy to improve the search efficiency. The experimental results validate the high efficiency and effectiveness of our proposed approach when solving various LSMOPs.

The maximum cut (MAX-CUT) problem is to find a bipartition of the vertices in a given graph such that the number of edges with ends in different sets reaches the largest. Though, several experimental investigations have shown that evolutionary algorithms (EAs) are efficient for this NP-complete problem, there is little theoretical work about EAs on the problem. In this paper, we theoretically investigate the performance of EAs on the MAX-CUT problem. We find that both the (1+1) EA and the (1+1) EA*, two simple EAs, efficiently achieve approximation solutions of (m/2)+(1/4)s(G) and (m/2)+(1/2)(√{8m+1}-1), where m and s(G) are respectively the number of edges and the number of odd degree vertices in the input graph. We also reveal that for a given integer k the (1+1) EA* finds a cut of size at least k in expected runtime O(nm+1/δ(4k)) and a cut of size at least (m/2)+k in expected runtime O(n(2)m+1/δ((64/3)k(2))), where δ is a constant mutation probability and n is the number of vertices in the input graph. Finally, we show that the (1+1) EA and the (1+1) EA* are better than some local search algorithms in one instance, and we also show that these two simple EAs may not be efficient in another instance.

This article critically investigates the limitations of the simulated annealing algorithm using probabilistic bits (pSA) in solving large-scale combinatorial optimization problems. The study begins with an in-depth analysis of the pSA process, focusing on the issues resulting from unexpected oscillations among p-bits. These oscillations hinder the energy reduction of the Ising model and thus obstruct the successful execution of pSA in complex tasks. Through detailed simulations, we unravel the root cause of this energy stagnation, identifying the feedback mechanism inherent to the pSA operation as the primary contributor to these disruptive oscillations. To address this challenge, we propose two novel algorithms, time average pSA (TApSA) and stalled pSA (SpSA). These algorithms are designed based on partial deactivation of p-bits and are thoroughly tested using Python simulations on maximum cut benchmarks that are typical combinatorial optimization problems. On the 16 benchmarks from 800 to 5000 nodes, the proposed methods improve the normalized cut value from 0.8 to 98.4% on average in comparison with the conventional pSA.

A novel approach using transient chaotic neural network (TCNN) and self-organizing feature map (SOFM) process to solve large-scale combinatorial optimization problems has been proposed. With the clustering function of self-organizing feature map, the computational cost of a large-scale combinatorial optimization problem solved by TCNN is reduced. Numerical simulation of TSP shows that the proposed method is effective to solve large-scale optimization problems.

Large-scale machine system optimization problems usually resist solution because large-scale machine systems often have a hierarchical general structure that produces numerous local optirna during hierarchical optimization. Genetic algorithms having hierarchical genotypes can be directly applied to the optimization of such problems, however the appearance of local optima during hierarchical optimization procedures often foils effective searching. To enable the global optimization of such problems, skillful analysis of the component relationships comprising the hierarchical problem allows the hierarchical structure to be divided into more tractable units. Divided optimization problems can then be optimized via multiple crenetic algorithm processes, and global large-scale optimization problems can be represented as hierarchical sets of genetic algorithm processes. The exchange of local searching information between these genetic algorithm processes enables global searching and parallel processing, both of which enhance the efficiency and quality of searching for optimum solutions to large-scale optimization problems.

We propose singular value decomposition (SVD) algorithms for very large-scale matrices based on a low-rank tensor decomposition technique called the tensor train (TT) format. By using the proposed algorithms, we can compute several dominant singular values and corresponding singular vectors of large-scale structured matrices given in a low-rank TT format. We propose a large-scale trace optimization problem, and in the proposed methods, the large-scale optimization problem is reduced to sequential small-scale optimization problems. We show that the computational complexity of the proposed algorithms scales logarithmically with the matrix size if the TT-ranks are bounded. Numerical simulations based on very large-scale Hilbert matrix demonstrate the effectiveness of the proposed methods.

Today, because of the exponential growth of the volume of data in big data applications, large-scale optimization problems (i.e., optimization problems with a large number of decision variables) have become ubiquitous in the real world [219, 220]. In this chapter, building on the algorithms and observations presented for solving both continuous and combinatorial optimization problems, we further present a novel multi-space evolutionary search framework based on EMT for solving large-scale single-objective optimization problem.

Earthquake rupture forecasting is a critical component of seismic hazard analysis and requires identifying physically plausible rupture paths across complex fault networks. At its core, this task could be formulated as a large-scale combinatorial optimization problem, which involves selecting an optimal subset of fault segments from a set of candidates. Such problems pose significant challenges for traditional algorithms, as the number of admissible multi-fault ruptures grows combinatorially. Current operational workflows rely on locally applied plausibility filters. While computationally efficient, such greedy local heuristics risk excluding globally competitive rupture scenarios, particularly in regions with dense fault connectivity and competing rupture pathways. This work investigates whether quantum annealing hardware can serve as a sampling accelerator for exploring the ensemble of physically plausible ruptures beyond what is currently accessible to classical approaches. Designing a problem formulation that remains scientifically meaningful while respecting the constraints of current quantum hardware (size, noise, etc.) is nontrivial, and naïve encodings often collapse under these limitations. We encode rupture plausibility modeling as low energy solutions to a quadratic unconstrained binary optimization (QUBO) problem defined on a fault-network middle graph. Binary variables represent activated fault segments, while local interaction terms encode Coulomb stress transfer (as a proxy for pairwise rupture compatibility), continuity preferences, and branching behavior. Here, we adopt a classical-quantum hybrid workflow: the quantum annealer is used to sample globally competitive rupture candidates, while classical post-processing implements physical constraints to filter out non-physical ruptures. The workflow is demonstrated on a subset of the fault network used in the 2023 USGS National Seismic Hazard Model including over 200 fault segments, using both simulated thermal and quantum annealing on D-Wave hardware. Results show that our hardware-aware formulation and conditioning enable robust sampling on comparatively large fault-network instances. Views on quantum computing are polarized: some overstate its power, while others dismiss it as impractical. Our results help bridge this by establishing a concrete, testable pathway for integrating quantum annealing into rupture-modeling workflows on existing purpose-built quantum devices.—---This work is supported by the ICSC National Research Centre for High Performance Computing, Big Data and Quantum Computing (CN00000013, CUP D53C22001300005) within the European Union-NextGenerationEU program (National Recovery and Resilience Plan (PNRR) - Mission 4 Component 2 Investment 1.4.)

Although there are many studies on large-scale unconstrained optimization (e.g., with 100 to 1000 variables) and small-scale constrained optimization (e.g., with 10 to 30 variables) using nature-inspired algorithms (e.g., evolutionary algorithms and swarm intelligence algorithms), no publicly available nature-inspired algorithm is developed for large-scale constrained optimization. In this paper, we combine a cooperative coevolutionary particle swarm optimization (CCPSO) algorithm with the e constrained method to solve large-scale real-valued constrained optimization problems. The eCCPSO framework is proposed, and three different algorithms based on the framework, i.e., eCCPSOd, eCCPSOw and eCCPSOw2, are developed. The proposed algorithms compare favorably to the state-of-the-art constrained optimization algorithm eDEag on large-scale problems. The experimental results further suggest that eCCPSOw2 with adaptive improvement detection technique is highly competitive compared with the other algorithms considered in this work for solving large-scale real-valued constrained optimization problems.

Although numbers of heuristic algorithms are successfully developed for solving portfolio optimization problems, this is not for all cases of the large-scale ones. A large-scale portfolio optimization involves dealing with the large search space and dense variance-covariance matrix associated with the problem. This paper proposed a new multi-objective algorithm for solving a large-scale optimization problem based upon the notion of cooperative coevolutionary algorithms (CCA). The new problem decomposition scheme was designed by allowing the species-size to be dynamically adjusted as the runs progress. This scheme enhances capability of traditional CCA in dealing with non-separable optimization problem. The collaborator selection method was modified to allow the proposed CCA to perform in a multi-objective (MO) optimization framework. Additionally, the proposed algorithm, named as “DMOCCA”, was implemented for solving large-scale portfolio optimization problem with cardinality constraint using the real-world data set having scale up to 2196 dimensions. Moreover, its performances were benchmarked with those of the SPEA-II and MOPSO.

This paper presents enhanced adaptive differential evolution (EADE) algorithm for solving high-dimensional optimization problems over continuous space. To utilize the information of good and bad vectors in the DE population, the proposed algorithm introduces a new mutation rule. It uses two random chosen vectors of the top and bottom 100p% individuals in the current population of size NP, while the third vector is selected randomly from the middle [NP-2(100p%)] individuals. The mutation rule is combined with the basic mutation strategy DE/rand/1/bin, where the only one of the two mutation rules is applied with the probability of 0.5. This new mutation scheme helps to maintain effectively the balance between the global exploration and local exploitation abilities for searching process of the DE. Furthermore, we propose a novel self-adaptive scheme for gradual change of the values of the crossover rate that can excellently benefit from the past experience of the individuals in the search space during evolution process which, in turn, can considerably balance the common trade-off between the population diversity and convergence speed. The proposed algorithm has been evaluated on the 7 and 20 standard high-dimensional benchmark numerical optimization problems for both the IEEE CEC-2008 and the IEEE CEC-2010 Special Session and Competition on Large-Scale Global Optimization. The comparison results between EADE and its version and the other state-of-art algorithms that were all tested on these test suites indicate that the proposed algorithm and its version are highly competitive algorithms for solving large-scale global optimization problems.

With the development of machine learning in various fields, it can also be applied to combinatorial optimization problems, automatically discovering generic and fast heuristic algorithms based on training data, and requires fewer theoretical and empirical knowledge. Pointer network improves the attention mechanism, instead of allocating different attention to hidden states of encoder to generate context vectors, using attention as a pointer to select an element of the input sequence at every step of decoding, which solves the problem of variable dictionary size of the output sequence. Pointer net (Ptr-Net) applied to three combinatorial optimization problems, convex hull, Delaunay triangulation, and traveling salesman problem (TSP), obtains good approximate solutions. Point matching is also a special kind of combinatorial optimization problems that is to obtain the optimal corresponding references, which can be modeled by Ptr-Net. However, Ptr-Net can’t be used to solve point matching problem because it doesn’t take full advantage of the correspondences between the two point sets. We propose multi-pointer network, which draws the idea from multi-label classification, to address this limitation by pointing out a set of input elements. These applications are all based on supervised learning to approximate expected known solutions. However, high-quality labeled data is often expensive, unreliable, or simply unavailable and may be infeasible for new problem statements, making supervised learning being unpractical. Reinforcement learning, as another research hotspot in the field of machine learning, does not require labeled sample data. It interacts with the environment through trial-and-error mechanism and focuses more on learning problem-solving strategies. We introduce a framework to tackle combinatorial optimization problems using neural networks and reinforcement learning, focusing on the traveling salesman problem. We also introduce a framework, a unique combination of reinforcement learning and graph embedding network, to solve graph optimization problems, focusing on maximum cut (MAXCUT) and minimum vertex cover (MVC) problems.

This paper addresses large-scale urban transportation optimization problems with time-dependent continuous decision variables, a stochastic simulation-based objective function, and general analytical differentiable constraints. We propose a metamodel approach to address, in a computationally efficient way, these large-scale dynamic simulation-based optimization problems. We formulate an analytical dynamic network model that is used as part of the metamodel. The network model formulation combines ideas from transient queueing theory and traffic flow theory. The model is formulated as a system of equations. The model complexity is linear in the number of road links and is independent of the link space capacities. This makes it a scalable model suitable for the analysis of large-scale problems. The proposed dynamic metamodel approach is used to address a time-dependent large-scale traffic signal control problem for the city of Lausanne. Its performance is compared to that of a stationary metamodel approach. The proposed approach outperforms the stationary approach. This comparison illustrates the added value of providing the algorithm with analytical dynamic problem-specific structural information. The performance of a signal plan derived by the proposed approach is also compared to that of an existing signal plan for the city of Lausanne, and to that of a signal plan derived by a mainstream commercial signal control software. The proposed method can systematically identify signal plans with good performance. The online appendix is available at https://doi.org/10.1287/trsc.2016.0717 .

Improving NSGA-III algorithms with information feedback models for large-scale many-objective optimization

We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had perfc~rmance guarantees of ~for MAX CUT and ~for MAX 2SAT. Slight extensions of our analysis lead to a .79607-approximation algorithm for the maximum directed cut problem (MAX DICUT) and a .758-approximation algorithm for MAX SAT, where the best previously known approxim ation algorithms had performance guarantees of ~and ~, respectively. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of :semidefinite programming in the design of approximation algorithms.