Unconstrained Optimization Research Articles

Energy minimization in charged polymer-multi-biomolecule electrolyte solution systems faces major challenges, where the energy landscape is typically highly nonconvex, ill-conditioned, and dominated by long-range electrostatic interactions. In such settings, standard nonlinear conjugate gradient (NCG) methods often struggle to maintain sufficient descent directions due to unstable conjugate gradient parameters caused by poor curvature information and frequent oscillations in gradient directions. To address this, we develop an enhanced NCG algorithm, termed the ELS (short for Enlong Shang) method, which introduces a modified conjugate gradient coefficient βkELS with a tunable denominator parameter ω, enabling improved stability in regions with poor local curvature. In addition, existing studies have shown that the convergence analysis of current NCG methods usually relies on the pre-setting of the parameter σ, whose theoretical bounds are difficult to adapt to the complex demands of high-dimensional nonconvex optimization problems. Hence, a novel convergence proof technique is proposed to show that the ELS method satisfies the sufficient descent condition for a broad range of line search parameters σ ∈ (0, 1), while still ensuring global convergence under nonconvex objectives. For traditional unconstrained optimization problems, the numerical performance of the ELS method outperforms the existing representative NCG methods. We apply it to the energy minimization phase in complex biomolecular simulations. Compared to direct dynamics simulation without preprocessing, implementing this minimization saves about 60% of the total time required to reach dynamic equilibrium, even exceeding the mainstream staged minimization strategy in Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). Importantly, the final conformation closely matches that of the purely dynamics simulation thermodynamically and has an acceptable energy deviation.

Directional Gradient and Curvature Approximation via Legendre Quadrature in Unconstrained Optimization

Large language models enhanced graph neural architecture search for quadratic unconstrained binary optimization

An efficient algorithm via a novel one-parameter filled function based on general univariate functions for unconstrained global optimization

A preprocessing technique for quadratic unconstrained binary optimization

The object of this study is multi-aircraft landing scheduling on single and multiple runways, which is an important aspect of modern air traffic management systems. The main problems solved in this research are the complexity of scheduling optimization due to limited runway capacity, the need to maintain a safe distance between aircraft, and the uncertainty of estimated time of arrival (ETA) which is often influenced by external factors such as weather and air traffic density. To overcome these challenges, this research proposes a hybrid approach between Long short-term memory-gradient boosting with the quantum annealing method. the results show that this approach is able to significantly improve the performance of the scheduling system, with an accuracy of 0.93, a precision of 0.91, a recall of 0.90, and an F1 score of 0.91. These values are higher than the model without quantum annealing, which only achieved an accuracy of 0.87, a precision of 0.85, a recall of 0.83, and an F1 score of 0.84. This improvement can be explained by the ability of LSTM-gradient boosting to predict ETA deviation more accurately, as well as the effectiveness of quantum annealing in solving the quadratic unconstrained binary optimization (QUBO) formulation efficiently. The unique feature of this research lies in the application of a hybrid model that combines the power of machine learning and quantum computing, achieving a balance between predictive accuracy and optimization efficiency. These research findings can be applied to air traffic scheduling systems at airports with single or multiple runways. Their implementation has the potential to improve operational efficiency, reduce delays, and enhance flight safety through more precise and adaptive landing time management

Ising Machines are emerging hardware architectures that efficiently solve NP-hard combinatorial optimization problems. Generally, combinatorial problems are transformed into quadratic unconstrained binary optimization (QUBO) form, but this transformation often complicates the solution landscape, degrading performance, especially for multi-state problems. To address this challenge, we model spin interactions as generalized boolean logic function to significantly reduce the exploration space. We demonstrate the effectiveness of our approach on graph coloring problem using probabilistic Ising solvers, achieving similar accuracy compared to state-of-the-art heuristics and machine learning algorithms. It also shows significant improvement over state-of-the-art QUBO-based Ising solvers, including probabilistic Ising and simulated bifurcation machines. We also design 1024-neuron all-to-all connected probabilistic Ising accelerator on FPGA with the proposed approach that shows sim10000times performance acceleration compared to GPU-based Tabucol heuristics and reducing physical neurons by 1.5-4times over baseline Ising frameworks. Thus, this work establishes superior efficiency, scalability and solution quality for multi-state optimization problems.

This study applies Harry Markowitz's Modern Portfolio Theory to analyze the portfolio optimization of five representative stocks (AAPL, JPM, JNJ, CVX, and KO) from different industries. The risk-return characteristics and actual performance are compared by constructing both unconstrained and constrained (long only) portfolios. The study is based on historical data from January 2022 to December 2024 for parameter estimation and an out-of-sample performance test is conducted during December 23-31, 2024 period. The study employs a mean-variance optimization framework with maximizing the Sharpe ratio as the objective function to solve the unconstrained and constrained optimization problems by Lagrange multiplier method and sequential quadratic programming algorithm, respectively. The results show that the constrained portfolio outperforms the unconstrained portfolio during the test period, achieving a total return of 2.38%, compared to the unconstrained portfolio, which only obtains a return of 1.44%. It is found that although unconstrained portfolios theoretically have higher expected Sharpe ratios (0.7739 vs. 0.5644), the constraints are effective in avoiding the risks associated with extreme weight allocations in practical applications. The unconstrained portfolio produces an extreme weight allocation (JPM weight 205.27%, JNJ weight 369.60%), while the maximum weight of the constrained portfolio is only 62.00%, with significant risk control. This study reveals the important difference between the theoretical optimum and the practical feasibility, proves the important role of appropriate constraints in the real investment environment, and provides valuable insights for investment practice.

Abstract Wireless sensor networks (WSNs) are increasingly being targeted by malicious users who abuse resource scarcity to mask attack origins, rendering traditional IP traceback methods inappropriate due to excessive latency, low accuracy, and unacceptable overhead. IP traceback in this paper is formulated as a quadratic unconstrained binary optimization (QUBO) problem, and quantum annealing is used via D-Wave hardware and hybrid solvers, and is integrated into an NS3 simulation environment. Comparative study with classical packet marking and probabilistic sampling regimes indicates that the quantum-enhanced model achieves 90% traceback success rates and 5–10 percentage point false positive reduction with comparable latency and energy expenses. These results affirm that cyberphysical and IoT domains can be significantly benefited by quantum annealing for attacker localization in WSNs, resulting in practical, low-overhead security solutions.

Winner-Take-All (WTA) is a foundational principle in modelling competitive dynamics, yet traditional neural network approaches to solving the WTA problem are often hampered by high computational complexity. This paper addresses this challenge by introducing a novel and computationally efficient dynamic competition model that circumvents the complexity of conventional constrained quadratic programming (QP) solvers. First, we reformulate the classical constrained WTA problem into an equivalent unconstrained optimisation problem. This is achieved by innovatively employing the Softmax function, which inherently satisfies the summation-to-one and non-negativity constraints. Subsequently, a simple yet powerful dynamic neural network is developed to solve this unconstrained problem, accompanied by rigorous theoretical proofs of its stability, global convergence, and state boundedness. Numerical experiments, conducted with both static and dynamic inputs, validate the model's efficacy. The results highlight the model's exceptional robustness, maintaining stable and accurate WTA selection even in the presence of significant Gaussian white noise, demonstrating a marked improvement over existing methods. The proposed framework is also readily adaptable to a broad range of WTA-related problems sharing similar structural characteristics, thereby offering substantial potential for diverse real-time decision-making applications.

In this paper, a new effective hybrid three-term conjugate gradient method with restart procedure is proposed to solve unconstrained optimizations. We propose a novel search direction that approximates the memoryless BFGS quasi-Newton direction, forming a hybrid structure derived from FR, CD, and DY conjugate parameters, which demonstrates excellent performance in large-scale problems. Its sufficient descent property is demonstrated. Under certain assumptions and the weak Wolfe line search conditions, the global convergence is analyzed. Two sets of numerical experiments on 100 test functions are conducted to evaluate the proposed algorithm. Numerical experiments show that it outperforms some other conjugate gradient algorithms. Furthermore, the proposed algorithm demonstrates superior performance in image restoration, achieving higher peak signal-to-noise ratio values.

With buildings accounting for 40% of global energy consumption, heating, ventilation, and air conditioning (HVAC) systems represent the single largest opportunity for emissions reduction, consuming up to 60% of commercial building energy while maintaining occupant comfort. This critical balance between energy efficiency and human comfort has traditionally relied on rule-based and model predictive control strategies. Given the multi-objective nature and complexity of modern HVAC systems, these approaches fall short in satisfying both objectives. Recently, reinforcement learning (RL) has emerged as a method capable of learning optimal control policies directly from system interactions without requiring explicit models. However, standard RL approaches frequently violate comfort constraints during exploration, making them unsuitable for real-world deployment where occupant comfort cannot be compromised. This paper addresses two fundamental challenges in HVAC control: the difficulty of constrained optimization in RL and the challenge of defining appropriate comfort constraints across diverse conditions. We adopt a safe RL with a neural barrier certificate framework that (1) transforms the constrained HVAC problem into an unconstrained optimization and (2) constructs these certificates in a data-driven manner using neural networks, adapting to building-specific comfort patterns without manual threshold setting. This approach enables the agent to almost guarantee solutions that improve energy efficiency and ensure defined comfort limits. We validate our approach through seven experiments spanning residential and commercial buildings, from single-zone heat pump control to five-zone variable air volume (VAV) systems. Our safe RL framework achieves energy reduction compared to baseline operation while maintaining higher comfort compliance than unconstrained RL. The data-driven barrier construction discovers building-specific comfort patterns, enabling context-aware optimization impossible with fixed thresholds. While neural approximation prevents absolute safety guarantees, reducing catastrophic safety failures compared to unconstrained RL while maintaining adaptability positions this approach as a developmental bridge between RL theory and real-world building automation, though the considerable gap in both safety and energy performance relative to rule-based control indicates the method requires substantial improvement for practical deployment.

Join order optimization is among the most crucial query optimization problems, and its central position is also evident in the new research field where quantum computing is applied to database optimization and data management. In this field, join order optimization is the most studied database problem, typically tackled with a quadratic unconstrained binary optimization model, which is solved using various meta-heuristics, such as quantum and digital annealing, the quantum approximate optimization algorithm, or the variational quantum eigensolver. In this study, we continue developing quantum computing techniques for left-deep join order optimization by presenting three novel quantum optimization algorithms. These algorithms are based on a higher-order unconstrained binary optimization model, which is a generalization of the quadratic model and has not previously been applied to database problems. Theoretically, these optimization problems naturally map to universal quantum computers and quantum annealers. Compared to previous studies, two of our algorithms are the first quantum algorithms to model the join order cost function precisely. We prove theoretical bounds by showing that these two methods encode the same plans as the dynamic programming algorithm with respect to the query graph, which provides the optimal result up to cross products. The third algorithm achieves plans at least as good as those of the greedy algorithm with respect to the query graph. These results establish a meaningful theoretical connection between classical and quantum algorithms for selecting left-deep join orders. To demonstrate the practical usability of our algorithms, we have conducted an extensive experimental evaluation on thousands of clique, cycle, star, tree, and chain query graphs using both quantum and classical solvers.

ABSTRACTApproximate block diagonalization is a problem of transforming a given symmetric matrix as close to block diagonal as possible by symmetric permutations of its rows and columns. This problem arises as a preprocessing stage of various scientific calculations and has been shown to be NP‐complete. In this paper, we consider solving this problem approximately using the D‐Wave Advantage quantum annealer. For this purpose, several steps are needed. First, we have to reformulate the problem as a quadratic unconstrained binary optimization (QUBO) problem. Second, the QUBO has to be embedded into the physical qubit network of the quantum annealer. Third, and optionally, reverse annealing for improving the solution can be applied. We propose two QUBO formulations and four embedding strategies for the problem and discuss their advantages and disadvantages. Through numerical experiments, it is shown that the combination of domain‐wall encoding and D‐Wave's automatic embedding is the most efficient in terms of usage of physical qubits, while the combination of one‐hot encoding and automatic embedding is superior in terms of the probability of obtaining a feasible solution. It is also shown that reverse annealing is effective in improving the solution for medium‐sized problems.

Abstract Physics-inspired computing paradigms are receiving renewed attention to enhance efficiency in compute-intensive tasks such as artificial intelligence and optimization. Similar to Hopfield neural networks, oscillatory neural networks (ONNs) minimize an Ising energy function that embeds the solutions of hard combinatorial optimization problems. Despite their success in solving unconstrained optimization problems, Ising machines still face challenges with constrained problems as they can become trapped in infeasible local minima. In this paper, we introduce a Lagrange ONN (LagONN) designed to escape infeasible states based on the theory of Lagrange multipliers. Unlike existing oscillatory Ising machines, LagONN employs additional Lagrange oscillators to guide the system towards feasible states in an augmented energy landscape, settling only when constraints are met. Taking the maximum satisfiability problem with three literals as a use case (Max-3-SAT), we harness LagONN's constraint satisfaction mechanism to find optimal solutions for random SATlib instances with up to 200 variables and 860 clauses, which provides a deterministic alternative to simulated annealing for coupled oscillators. We benchmark LagONN with SAT solvers and further discuss the potential of Lagrange oscillators to address other constraints, such as phase copying, which is useful in oscillatory Ising machines with limited connectivity.

Steric clashes pose a challenge when exploring dense protein systems using conventional explicit-chain methods. A minimal example is a single lattice protein confined on a minimal grid, with no free sites. Finding its minimum energy is a hard optimization problem, with similarities to scheduling problems. It can be recast as a quadratic unconstrained binary optimization (QUBO) problem amenable to classical and quantum approaches. We show that this problem in its QUBO form can be swiftly and consistently solved for chain length 48, using either classical simulated annealing or hybrid quantum-classical annealing on a D-Wave system. In fact, the latter computations required about 10 s. We also test linear and quadratic programming methods, which work well for a lattice gas but struggle with chain constraints. All methods are benchmarked against exact results obtained from exhaustive structure enumeration, at a high computational cost.

The filled function is considered to be an effective method for solving global optimization problems. It obtains the global optimal solution of the optimization problem by alternating the two stages of minimization and filling. Firstly, this paper proposes a new parameter-free and continuously differentiable filled function, which does not contain exponential terms and logarithmic terms, and overcomes some shortcomings in the form of the original parameter filled function. Secondly, it is proved that the proposed function satisfies the filled property and some good analytical properties, and the corresponding filled function algorithm is given. Experiments are carried out on some benchmark functions and the application of earthwork allocation problem. Finally, the numerical results show the effectiveness and stability of the algorithm.

Enhanced open-source scatter search algorithm for solving quadratic unconstrained binary optimization problems

This study introduces a new and efficient modification of the conjugate gradient algorithm for solving non-convex unconstrained optimization problems. The proposed method ensures the sufficient descent property regardless of the line search technique and is proven to be globally convergent under both Wolfe and Armijo conditions. Its numerical performance is assessed through a set of large-scale benchmark problems. The findings indicate that the proposed algorithm exhibits competitive efficiency and reliability compared to existing conjugate gradient variants. To demonstrate applicability further, the algorithm is tested on two scenarios. The first is an image restoration problem, and the second is the motion control of a 2-DOF planar robotic manipulator, where inverse kinematics is solved iteratively for trajectory tracking. The algorithm demonstrates high tracking precision and stable convergence, highlighting its theoretical soundness and potential for various optimization applications.

Transformation of Bi-level Transit Network Design Problem into Single-objective Unconstrained Optimization