Unconstrained Optimization Problems Research Articles

Solving QUBO problems with cP systems

AbstractP systems with compound terms (cP systems) have been proposed by Radu Nicolescu in 2018. These expressive cP systems have been used to solve well-known NP-complete problems efficiently, such as the Hamiltonian path, traveling salesman, 3-coloring, and software verification problems. In this paper, we use cP systems to provide an efficient parallel solution to the integer-valued quadratic unconstrained Boolean optimization (QUBO) problem.

A novel metaheuristic based on object-oriented programming concepts for engineering optimization

This paper presents a novel, robust, efficient, and simple optimization algorithm called the Object-Oriented Programming Optimization Algorithm (OOPOA) for tackling constrained and unconstrained optimization problems. The algorithm is inspired by the inheritance concept of Object-Oriented programming languages, where the features of a class are classified into three types according to inheritance probability: public, private, and protected. The object-oriented programming inheritance concept is implemented in the algorithm to update the population for the next generations. The proposed algorithm ensures exploitation by selecting the solution with the highest fitness to be inherited in each iteration. It ensures exploration by applying a mutation process that helps explore wide regions in the search space. The performance of this technique is demonstrated by solving 34 different optimization tasks, including 20 standard benchmark problems, ten IEEE Congress of Evolutionary Computation benchmark test functions, and four constrained real-world engineering design problems.

Another modified version of RMIL conjugate gradient method

Due to their simplicity and global convergence properties, the conjugate gradient (CG) methods are widely used for solving unconstrained optimization problems, especially those of large scale. To establish the global convergence and to obtain better numerical performance in practice, much effort has been devoted to develop new CG methods or even to modify well- known methods. In 2012, Rivaie et al., have proposed a new CG method, called RMIL which has good numerical results and globally convergent under the exact line search. However, in 2016, Dai has pointed out a mistake in the steps of the proof of global convergence of RMIL and hence to guarantee the global convergence he suggested a modified version of RMIL, called RMIL+. In this paper, we present another modified version of RMIL, which is globally convergent via the exact line search. Furthermore, to support the theoretical proof of the global convergence of the modified version in practical computation, a numerical experiment based on comparing it with RMIL, RMIL+, and CG-DESCENT was done.

Capacitated Clustering via Majorization-Minimization and Collaborative Neurodynamic Optimization.

This paper addresses capacitated clustering based on majorization-minimization and collaborative neurodynamic optimization (CNO). Capacitated clustering is formulated as a combinatorial optimization problem. Its objective function consists of fractional terms with intra-cluster similarities in their numerators and cluster cardinalities in their denominators as normalized cluster compactness measures. To obviate the difficulty in optimizing the objective function with factional terms, the combinatorial optimization problem is reformulated as an iteratively reweighted quadratic unconstrained binary optimization problem with a surrogate function and a penalty function in a majorization-minimization framework. A clustering algorithm is developed based on CNO for solving the reformulated problem. It employs multiple Boltzmann machines operating concurrently for local searches and a particle swarm optimization rule for repositioning neuronal states upon their local convergence. Experimental results on ten benchmark datasets are elaborated to demonstrate the superior clustering performance of the proposed approaches against seven baseline algorithms in terms of 21 internal cluster validity criteria.

A time-fuel hybrid optimal guidance for micro/nano satellite’s glideslope approach

Aimed at the trajectory planning problem of micro/nano satellites’ approaching missions under multiple constraints, this paper proposes a V-bar glideslope maneuver optimization method considering both time and fuel consumption. A dual-objective trajectory optimization model is established based on the CW motion equation, considering the constraints of trajectory and impulse. The optimal solution set of the glideslope point position and maneuvering time is obtained by employing the NSGA2 algorithm. In order to reduce computational complexity, a method for simplifying trajectory optimization is proposed. By solving a constrained dual-impulse optimization problem, the feasible range of distance and transfer time between adjacent glideslope points are obtained, which transforms the constraint equations of the glideslope maneuver optimization into the search space of optimization variables. The problem is simplified to an unconstrained optimization problem. Finally, a simulation is conducted with the approaching mission of a typical micro/nano satellite. The results demonstrate that the proposed method can provide an optimal solution set considering both fuel and time consumption, offering a flexible maneuver planning strategy for on-orbit application.

Improved FPA for aircraft conceptual design

Aircraft weight design is a key technology in aircraft conceptual design. The design of aircraft weight is directly related to many design parameters of the aircraft, and there are complex constraint relationships between these parameters. It is difficult to determine the design parameters in the process of aircraft conceptual design with an accurate mathematical analytical model, it is necessary to use an optimization algorithm. Firstly, to build the optimization object function, the aircraft weight function and some constrain functions are given out, thus, the aircraft weight design problem is transformed into a single objective multi-constraint optimization problem, aiming at the problem of parameter coupling, Aitken acceleration algorithm is proposed to solve the problem through iterative, thus, and a feasible solution of aircraft parameters are obtained; Secondly, considering the advantages of flower pollination algorithms in solving unconstrained problems, the flower pollination algorithm (FPA) combined with the Aitken acceleration algorithm is proposed for automatic global optimization, to avoid the FPA falling into the local optimal solution or missing optimal solution, propose improvements to the FPA algorithm from three aspects: adaptive conversion probability, adaptive step size adjustment, and adaptive reproduction probability generation. To use FPA, the multi-constraint optimization problem is transformed into a multi-parameter unconstrained optimization problem with a penalty term based on the penalty function method. Finally, a design example is given to verify the effectiveness of the algorithm, and the superiority of the improved algorithm is verified by comparing it with PSO and the fixed transition probability FPA.

Reliability control of nonlinear stochastic dynamical system based on discrete data

Stochastic optimal control based on reliability plays a crucial role in mitigating structural failure and ensuring safe operation. For general stochastic vibration systems, a data-driven method for reliability-based optimal control is proposed based on random state data. Firstly, the feedback control is split into conservative and dissipative components in coherence with physical intuition, and each component is expanded upon using pre-selected basic functions, resulting in a modified system with undetermined coefficients. Then, from discrete random samples, the expressions for the probability densities of the first-passage time and the stationary response can be identified, explicitly including system and excitation parameters, as well as initial and safety boundary conditions. Finally, a performance index, combining system reliability and control cost, is constructed and can be explicitly expressed in terms of the undetermined coefficients of the control forces. The optimal control problem for determining the optimal control forces is reformulated as an optimization problem for determining the coefficients that minimize the performance index. This unconstrained optimization problem can be easily solved. Three typical nonlinear stochastic systems, a controlled Duffing oscillator, a controlled nonlinear hysteretic system, and a controlled 2-DOF nonlinear dynamical system, are given to illustrate the validity and accuracy of the proposed data-driven method. Additionally, the control performance of the optimal control strategy in improving the system reliability is also discussed.

QUBO Problem Formulation of Fragment-Based Protein-Ligand Flexible Docking.

Protein-ligand docking plays a significant role in structure-based drug discovery. This methodology aims to estimate the binding mode and binding free energy between the drug-targeted protein and candidate chemical compounds, utilizing protein tertiary structure information. Reformulation of this docking as a quadratic unconstrained binary optimization (QUBO) problem to obtain solutions via quantum annealing has been attempted. However, previous studies did not consider the internal degrees of freedom of the compound that is mandatory and essential. In this study, we formulated fragment-based protein-ligand flexible docking, considering the internal degrees of freedom of the compound by focusing on fragments (rigid chemical substructures of compounds) as a QUBO problem. We introduced four factors essential for fragment-based docking in the Hamiltonian: (1) interaction energy between the target protein and each fragment, (2) clashes between fragments, (3) covalent bonds between fragments, and (4) the constraint that each fragment of the compound is selected for a single placement. We also implemented a proof-of-concept system and conducted redocking for the protein-compound complex structure of Aldose reductase (a drug target protein) using SQBM+, which is a simulated quantum annealer. The predicted binding pose reconstructed from the best solution was near-native (RMSD = 1.26 Å), which can be further improved (RMSD = 0.27 Å) using conventional energy minimization. The results indicate the validity of our QUBO problem formulation.

Binary matrix factorization via collaborative neurodynamic optimization

Binary matrix factorization is an important tool for dimension reduction for high-dimensional datasets with binary attributes and has been successfully applied in numerous areas. This paper presents a collaborative neurodynamic optimization approach to binary matrix factorization based on the original combinatorial optimization problem formulation and quadratic unconstrained binary optimization problem reformulations. The proposed approach employs multiple discrete Hopfield networks operating concurrently in search of local optima. In addition, a particle swarm optimization rule is used to reinitialize neuronal states iteratively to escape from local minima toward better ones. Experimental results on eight benchmark datasets are elaborated to demonstrate the superior performance of the proposed approach against six baseline algorithms in terms of factorization error. Additionally, the viability of the proposed approach is demonstrated for pattern discovery on three datasets.

Joint Hybrid Beamforming Design for Millimeter Wave Amplify-and-Forward Relay Communication Systems

Hybrid beamforming (HBF) has been regarded as one of the most promising technologies in millimeter Wave (mmWave) communication systems. In order to guarantee the communication quality in non-line-of-sight (NLOS) scenarios, joint HBF design for the mmWave amplify-and-forward (AF) relay communication system is studied in this paper. The ideal case is first considered where the mmWave half-duplex (HD) AF relay system operates with channel state information (CSI) accurately known. In order to tackle the non-convex problem, a manifold optimization (MO)-based alternating optimization algorithm is proposed, where an optimization problem containing only constant modulus constraints in Euclidean space can be converted to an unconstrained optimization problem in a Riemann manifold. Furthermore, considering more practical cases with estimation errors of CSI, we investigate the robust joint HBF design with the system operating in full-duplex (FD) mode to obtain higher spectral efficiency (SE). A null-space projection (NP) based self-interference cancellation (SIC) algorithm is developed to attenuate the self-interference (SI). Different from the traditional SI suppression algorithm, there’s no limit on the number of RF chains. Numerical results reveal that our proposed algorithms has a good convergence and can effectively deal with the influence of different CSI estimation errors. A significant performance improvement can be achieved in contrast with other approaches.

Neural network based approach for solving problems in plane wave duct acoustics

Neural networks have emerged as a tool for solving differential equations in many branches of engineering and science. But their progress in frequency domain acoustics is limited by the vanishing gradient problem that occurs at higher frequencies. This paper discusses a formulation that can address this issue. The problem of solving the governing differential equation along with the boundary conditions is posed as an unconstrained optimization problem. The acoustic field is approximated to the output of a neural network which is constructed in such a way that it always satisfies the boundary conditions. The applicability of the formulation is demonstrated on popular problems in plane wave acoustic theory. The predicted solution from the neural network formulation is compared with those obtained from the analytical solution. A good agreement is observed between the two solutions. The method of transfer learning to calculate the particle velocity from the existing acoustic pressure field is demonstrated with and without mean flow effects. The sensitivity of the training process to the choice of the activation function and the number of collocation points is studied.

An efficient trust region algorithm with bounded iteration sequence for unconstrained optimization and its application in support vector machine

In this paper, we propose an efficient trust region algorithm for unconstrained optimization problem. Different from the traditional trust region algorithm, our algorithm obtains the next iteration point by projection after getting the trust region trial step. The main feature of this algorithm is that it generates a bounded iteration sequence automatically, and it retains the excellent convergence properties. The numerical tests for support vector machine(SVM) and some unconstrained optimization test sets problems show the efficiency of the proposed algorithm.

Multi-Objective Portfolio Optimization Using a Quantum Annealer

In this study, the portfolio optimization problem is explored, using a combination of classical and quantum computing techniques. The portfolio optimization problem with specific objectives or constraints is often a quadratic optimization problem, due to the quadratic nature of, for example, risk measures. Quantum computing is a promising solution for quadratic optimization problems, as it can leverage quantum annealing and quantum approximate optimization algorithms, which are expected to tackle these problems more efficiently. Quantum computing takes advantage of quantum phenomena like superposition and entanglement. In this paper, a specific problem is introduced, where a portfolio of loans need to be optimized for 2030, considering ‘Return on Capital’ and ‘Concentration Risk’ objectives, as well as a carbon footprint constraint. This paper introduces the formulation of the problem and how it can be optimized using quantum computing, using a reformulation of the problem as a quadratic unconstrained binary optimization (QUBO) problem. Two QUBO formulations are presented, each addressing different aspects of the problem. The QUBO formulation succeeded in finding solutions that met the emission constraint, although classical simulated annealing still outperformed quantum annealing in solving this QUBO, in terms of solutions close to the Pareto frontier. Overall, this paper provides insights into how quantum computing can address complex optimization problems in the financial sector. It also highlights the potential of quantum computing for providing more efficient and robust solutions for portfolio management.

SOLVING FUZZY NONLINEAR PROGRAMMING PROBLEMS WITH EXTERIOR PENALTY FUZZY VALUED FUNCTIONS

In this article, we explore the exterior penalty fuzzy valued function method for Fuzzy Nonlinear Programming Problems (FNLPP). We present methods of penalty fuzzy valued function for solving constrained optimization problems by converting to unconstrained optimization problems. Triangular fuzzy numbers are used to represent the problem’s decision variables and coefficients. Using a new fuzzy arithmetic and fuzzy ranking method on the parametric form of triangular fuzzy numbers, the optimal solution of the FNLPP is found by applying the exterior penalty fuzzy valued function method without changing to its equivalent crisp form. We present a numerical example of the suggested method and compare the results to those produced by existing methods

Qubit Efficient Quantum Algorithms for the Vehicle Routing Problem on Noisy Intermediate‐Scale Quantum Processors

AbstractThe vehicle routing problem with time windows (VRPTW) is a common optimization problem faced within the logistics industry. In this work, the use of a previously‐introduced qubit encoding scheme is explored to reduce the number of qubits, to evaluate the effectiveness of Noisy Intermediate‐Scale Quantum (NISQ) devices when applied to industry relevant optimization problems. A quantum variational approach is applied to a testbed of multiple VRPTW instances ranging from 11 to 3964 routes. These intances are formulated as quadratic unconstrained binary optimization (QUBO) problems based on realistic shipping scenarios. The results are compared with standard binary‐to‐qubit mappings after executing on simulators as well as various quantum hardware platforms, including IBMQ, AWS (Rigetti), and IonQ. These results are benchmarked against the classical solver, Gurobi. The approach can find approximate solutions to the VRPTW comparable to those obtained from quantum algorithms using the full encoding, despite the reduction in qubits required. These results suggest that using the encoding scheme to fit larger problem sizes into fewer qubits is a promising step in using NISQ devices to find approximate solutions for industry‐based optimization problems, although additional resources are still required to eke out the performance from larger problem sizes.

A modified two-parameter scaled Broyden-type algorithm for unconstrained optimization problems

The Broyden-type algorithm is one of the most effective quasi-Newton methods for solving unconstrained optimization problems. However, the study of the global convergence of the Broyden-type algorithm is not enough. Motivated by Andrei (J. Comput. Appl. Math. 332: 26–44, 2018), we present a modified two-parameter scaled Broyden-type algorithm. Some approaches are used in the designed method: (1) we introduce a modified Broyden-type formula with two parameters; (2) we use approximated parameter settings to reduce the effect of eigenvalues of sequence {Bk}; (3) we propose a modified weak Wolfe–Powell line search technique with projection technique to avoid using failed search direction. The parameters of our study have the same functionality, that of correcting the structure of the eigenvalues of the approximate Hessian matrix. The difference is in the approach of 4-term Broyden updates instead of BFGS. Numerical performance shows that the modified Broyden-type algorithm is competitive with the classical Broyden approach.

Stabilized BB projection algorithm for large-scale convex constrained nonlinear monotone equations to signal and image processing problems

The Barzilai–Borwein (BB) method is a popular and efficient gradient method for solving large-scale unconstrained optimization problems. In general, it converges much faster than the steepest descent (Cauchy) method. However, it may not converge, even when the objective function is strongly convex. To overcome this, by virtue of bounding the distance between sequential iterates, [Burdakov et al. (2019)] proposed a stabilized BB method. In this paper, by combining the stabilized BB method with a projection approach, we develop a stabilized BB projection (SBBP) method to solve nonlinear monotone equations with convex constraints. SBBP does not involve computing matrices proving the usefulness for solving large-scale problems. Its global convergence and Q-linear convergence rate are also established. We report numerical experiments on recovering sparse signals and restoring blurred images arising from compressive sensing, demonstrating the usefulness of SBBP.

Integration of Modified Classical Conjugate Gradient Methods for Unconstrained Optimization

The integration of modified classical conjugate gradient methods (CGMs) for unconstrained optimization represents a crucial and evolving area of research within the field of optimization algorithms. Over time, numerous studies have put forth diverse modifications and novel approaches to enhance the effectiveness of classical CGMs. These modifications aim to address specific challenges and improve the overall performance of optimization algorithms in unconstrained scenarios. In order to tackle unconstrained optimization challenges and improve our understanding of their synergies, this ongoing study aims to unify different modified classical CGMs. Conventional CGMs have proven effective for optimization tasks, and a range of different approaches have been produced by carefully modifying these techniques. The main goal of this paper is to combine these modified versions, with particular attention to those that have similar numerators. The integration process involves systematically merging the advantageous aspects of these modified methods to develop not only innovative but also more resilient approaches to unconstrained optimization problems. The ultimate goal of this unification effort is to capitalize on the strengths inherent in different approaches to create a cohesive framework that significantly improves overall optimization performance. To thoroughly assess the efficacy of the integrated methods, a series of comprehensive performance tests are conducted. These tests include a meticulous comparison of outcomes with those of classical CGMs, providing valuable insights into the relative strengths and weaknesses of the modified approaches across diverse optimization scenarios. The evaluation criteria encompass convergence rates, solution accuracy, and computational efficiency. The conclusive outcome demonstrates that the unified approaches consistently outperform individual methods across all three crucial evaluation criteria.

A family of limited memory three term conjugate gradient methods

Based on several modified Hestenes–Stiefel and Polak–Ribière–Polyak nonlinear conjugate gradient methods, a family of three term limited memory CG methods are developed. When the current search direction falls into the subspace spanned by the previous m directions, the algorithm branches to optimize the objective function on the subspace by using L-BFGS method. We use this strategy to avoid potential local loops and accelerate the convergence. The proposed methods are sufficient descent. When the steplength is determined by Wolfe or Armijo line search, we establish the global convergence of the methods in a concise way. Numerical results verify the efficiency of the proposed method for the unconstrained optimization problems in the CUTEst library.

Digital IIR filters: Effective in edge preservation?

Infinite Impulse Response (IIR) filtering stands as a fundamental technique in various signal processing applications. Despite its widespread use, the scientific community widely acknowledges the suboptimal nature of IIR filters in preserving edges or abrupt transitions in piecewise signals. In situations where meticulous preservation of abrupt signal transitions is crucial, signal processing practitioners often explore alternative methodologies tailored to specific application demands. Consequently, the effectiveness of digital IIR filters in preserving signal edges while filtering out undesirable noise components continues to be a topic of active debate.In response to this limitation (or answering the question posed by the above title), the paper demonstrates that the suboptimal nature of IIR filters in preserving edges arises from the conventional methodology used to compute the output of zero-phase IIR filters. The standard method of output computation equates to solving an unconstrained optimization problem with an ℓ2-norm penalty term. Following this analysis, the paper introduces a refined approach to compute the output of digital IIR filters, aiming to overcome the challenge of preserving abrupt signal transitions. This innovative proposal replaces the ℓ2-norm with an ℓ1-norm, resulting in the development of an IIR filter that dynamically adjusts its coefficients based on the input signal. As a result, this novel approach to output computation effectively preserves essential signal features while simultaneously reducing noise, highlighting the efficacy of digital IIR filters in edge preservation. Therefore, the answer to the aforementioned question greatly depends on the chosen method for computing the output. By employing the suggested approach, the answer is unequivocally yes.