Unconstrained Problems Research Articles

A Descent Matrix-free Nonlinear Conjugate Gradient Algorithm for Impulse Noise Removal

The convergence of the Polak, Ribie‘re-Polyak (PRP) conjugate gradient (CG) method requires some modifications for improved theoretical properties. In this article, we explore an optimal choice of the Perry conjugacy condition to propose a hybrid CG parameter for solving optimization and inverse problems, particularly in an image reconstruction model. This parameter is selected to satisfy a combination of revised version of the PRP and Dai-Yuan (DY) CG methods. The numerical implementation includes inexact line search, showcasing the scheme's robustness (highest number of solved functions) compared to other known CG algorithms. The efficiency is shown in terms of Real error (RelErr), peak signal noise ratio (PNSR), and CPU time in seconds for impulse noise removal while for unconstrained minimization problems, the study evaluated the efficiency based on number of iterations, function evaluation, and CPU time in seconds. An interesting feature of the proposed method is its ability to converges to the minimizer regardless of the initial guess, relying on certain established assumptions.

Performance analyses of weighted superposition attraction-repulsion algorithms in solving difficult optimization problems

ABSTRACT The purpose of this paper is to test the performance of the recently proposed weighted superposition attraction-repulsion algorithms (WSA and WSAR) on unconstrained continuous optimization test problems and constrained optimization problems. WSAR is a successor of weighted superposition attraction algorithm (WSA). WSAR is established upon the superposition principle from physics and mimics attractive and repulsive movements of solution agents (vectors). Differently from the WSA, WSAR also considers repulsive movements with updated solution move equations. WSAR requires very few algorithm-specific parameters to be set and has good convergence and searching capability. Through extensive computational tests on many benchmark problems including CEC’2015 and CEC’2020 performance of the WSAR is compared against WSA and other metaheuristic algorithms. It is statistically shown that the WSAR algorithm is able to produce good and competitive results in comparison to its predecessor WSA and other metaheuristic algorithms.

Fast generation of collision-free low-thrust trajectories for asteroid landing using deep neural networks

This study presents a fast generation method of time-optimal low-thrust trajectories based on deep neural networks (DNN) for the collision-free asteroid landings. An equivalent unconstrained differentiable problem is transformed via the introduction of smooth penalty function, so that it can be addressed with the optimal control framework. To efficiently solve the equivalent problem to generate sufficient dataset for training DNNs, a double homotopy method is developed. Specifically, the equivalent problem is connected to the time-optimal control problem without anti-collision through two consecutive simplifications, and a solving process with double backward continuation is also formulated. Besides, two DNN models are constructed to provide high-quality predictions for the minimum glide-slope constraint angle of an initial state, and the initial costates and landing time for time-optimal low-thrust trajectories with anti-collision constraints, respectively. As such, the efficiency of collision-free low-thrust landing trajectory generation is improved significantly. For the cases within the range of training dataset or within a certain extension, the low-thrust landing trajectories with a very small terminal landing error can be generated using the predicted initial costates. Meanwhile, with the high-quality initial costates, the high-precision landing trajectories can be obtained by the double homotopy method in only a few iterative steps. Finally, the promising results in a Bennu collision-free asteroid landing scenario validate the efficiency of the proposed method.

EXPRESS: Constrained Assortment Optimization under the Cross-Nested Logit Model

We study the assortment optimization problem under general linear constraints, where the customer choice behavior is captured by the Cross-Nested Logit model. In this problem, there is a set of products organized into multiple subsets (or nests), where each product can belong to more than one nest. The aim is to find an assortment to offer to customers so that the expected revenue is maximized. We show that, under the Cross-Nested Logit model, the unconstrained assortment problem is NP-hard even when there are only two nests, and the problem is generally NP-hard to approximate to any constant factors. To tackle this challenging problem, we develop a new discretization mechanism to approximate the problem by a linear fractional program with a performance guarantee of [Formula: see text], for any accuracy level ε > 0. We then show that optimal solutions to the approximate problem can be obtained by solving mixed-integer linear programs. We further show that our discretization approach can also be applied to solve a joint assortment optimization and pricing problem, as well as an assortment problem under a mixture of Cross-Nested Logit models to account for multiple classes of customers. Our empirical results on a large number of randomly generated test instances demonstrate that, under a performance guarantee of 90% (i.e., expected revenues are guaranteed to be at least 90% of the optimal revenue), the percentage gaps between the objective values obtained from our approximation methods and the optimal expected revenues are no larger than 1.2%.

Exploring Enhanced Conjugate Gradient Methods: A Novel Family of Techniques for Efficient Unconstrained Minimization

Given that the conjugate gradient method (CGM) is computationally efficient and user-friendly, it is often used to address large-scale, unconstrained minimization issues. Numerous researchers have created new conjugate gradient (CG) update parameters by modifying the initial set, also referred to as classical CGMs. This has resulted in the development of several hybrid approaches. This work's major goal is to create a new family of techniques that can be used to create even more new methods. Consequently, Hestenes-Stiefel's update parameter and a new family involving Polak-Ribiere-Polyak and Liu-Storey CGMs are considered. By changing the parameters of this CGM family, a novel approach that possesses sufficient descent characteristics is obtained. A numerical experiment including many unconstrained minimization problems (UMP) is carried out to assess the novel method's efficacy compared to existing approaches. The result reveals that the new CG approach performs better than the current ones.

Massive MIMO secure beamforming design via manifold optimization combined with momentum

Secure beamforming with constant modulus has been regarded as a promising solution in enhancing the physical-layer security by intelligently designing the antenna phase of massive multiple-input multiple-output (MIMO) devices. The resulting problem is non-convex and NP-hard due to the constant modulus constraint (CMC). Existing methods mainly solve the problem by relaxing the constraints or the objective function, which inevitably introduces the relaxation error. We notice that the complex circle manifold (CCM), where complex numbers maintain a constant modulus, naturally satisfies the CMC. Further, considering the iteration points as high-dimensional particle points, the optimization process on the CCM can be regarded as the motion of the particle on the high-dimensional plane formed by the manifold. Based on this feature, we propose the manifold optimization method combining momentum (MOCM). Concretely, the original problem is first transformed into an unconstrained problem on the complex circle manifold. Then, the descent direction is derived to solve the problem by incorporating momentum into gradient information, which speeds up the convergence process. Simulation results show that the proposed method is 1 bps/Hz higher than existing works.

Optimal Uniform Strength Design of Frame and Lattice Structures

This paper provides a procedure to obtain the uniform strength of frame and lattice structures. Uniform strength condition is achieved by performing the shape optimization of all beam elements of the structure. The beam shape which guarantees uniform strength is analytically deduced from the one-dimensional Timoshenko model. The optimization problem presents itself as the search for the zeros of the objective-functions vector, which is a non-linear system of equations representing the kinematic-congruence and forces balance at every node of the structure. The analytical formulation of the optimization problem allows to construct the objective-functions vector without the use of external structural computation, i.e. not recurring to any Finite Element Analysis to accomplish iterations. This latter feature entails a great advantage in terms of computing time required to perform optimization. The proposed analytical formulation allows to directly insert the uniform strength condition into the objective-functions vector, transforming the optimization into an unconstrained problem. Some examples are shown in which the performance of the optimization procedure is discussed in terms of robustness and rate of computational complexity while increasing the degrees of freedom of the structure. The reliability and the quality of the optimization are verified through Finite Element Analysis.

Determining the optimal dome shape using a novel variable slippage coefficient non‐geodesic

AbstractThe research of high‐performance composite pressure vessels in practical application has become the focus of attention, and fiber paths play an important role in the structural performance of composite pressure vessels. Therefore, a method is proposed for determining the optimal geometric parameters of the pressure vessel head shape under a novel variable slippage coefficient non‐geodesic(NVSCNG). First, NVSCNG is used as the dome fiber path design mode, and the classical lamination theory is used to construct a dome stress field model. In addition, on the basis of the Tsai‐Wu failure criterion, the minimum laminate thickness is determined to satisfy the strength constraints. Second, the performance factor (pressure*volume/weight) is used to calculate the structural performance of the dome with variable geometric parameters under the novel variable slippage coefficient fiber path. Moreover, the stability constraints are considered to determine the optimal geometric parameter values of the dome. Finally, the structural properties, stress response and thickness were analyzed under two types of fiber paths based on the optimal dome profile. Results show that compared to the conventional variable slippage coefficient, the performance factor can be maximally improved by nearly 60.94% using the novel variable slippage coefficient, and the thickness of the composite layer at the equator of the dome can be maximally reduced by nearly 34.88%. In addition, the research shows that NVSCNG can improve the structural performance of the dome and realize the lightweight of dome components, which are greatly important in guiding the design of fiber paths for pressure vessels.Highlights No unconstrained linear problem between NVSCNG and initial winding angle. Optimal dome profile geometry parameters are m = 0.8, rc = 0.5. The dome structural failure is controlled by the longitudinal stresses. Winding angle distribution has a strong influence on dome stress distribution. Higher dimensionless performance factor and lighter mass for NVSCNG.

Introducing TTMDY, a three-term modified DY conjugate gradient direction for large-scale unconstrained problems

The main focus of our paper is a novel approach to enhance the MDY conjugate gradient direction. The key modification involves incorporating a third term, which plays a crucial role in determining the descent direction. By introducing this additional term, we transform the MDY conjugate gradient direction into a three-term conjugate gradient direction. This modification aims to improve the convergence properties of the algorithm and enhance its performance in solving optimization problems. In comparison to traditional MDY conjugate gradient methods, our approach demon- strates improved convergence properties and achieves higher solution quality. Numer- ical results confirm the superiority of our proposed method in terms of optimization performance. This highlights the potential of our modified approach to effectively tackle a wide range of optimization problems in various domains. The results of our numerical experiments provide strong evidence of the efficacy of our modified three- term conjugate gradient direction. В этой статье основное внимание уделяется представлению нового подхода к улучшению метода сопряженного градиента MDY. Cущественная модификация предполагает включение третьего члена, который играет решающую роль в определении направления спуска. Вводя этот дополнительный член, мы преобразуем направление сопряженного градиента MDY в трехчленное направление сопряженного градиента. Целью данной модификации является улучшение свойств сходимости алгоритма и повышение его производительности при решении оптимизационных задач. По сравнению с традиционными методами сопряженных градиентов MDY наш подход демонстрирует улучшенные свойства сходимости и обеспечивает более высокое качество решения. Численные результаты подтверждают превосходство предложенного метода с точки зрения производительности оптимизации. Это подчеркивает потенциал предложенного модифицированного подхода для эффективного решения широкого спектра задач оптимизации в различных областях. Результаты численных экспериментов убедительно свидетельствуют об эффективности метода модифицированного направления трехчленного сопряженного градиента.

Some convergently three-term trust region conjugate gradient algorithms under gradient function non-Lipschitz continuity

This paper introduces two three-term trust region conjugate gradient algorithms, TT-TR-WP and TT-TR-CG, which are capable of converging under non-Lipschitz continuous gradient functions without any additional conditions. These algorithms possess sufficient descent and trust region properties, and demonstrate global convergence. In order to assess their numerical performance, we compare them with two classical algorithms in terms of restoring noisy gray-scale and color images as well as solving large-scale unconstrained problems. In restoring noisy gray-scale images, we set the performance of TT-TR-WP as the standard, then TT-TR-CG takes around 2.33 times longer. The other algorithms around 2.46 and 2.41 times longer, respectively. In solving the same color images, the proposed algorithms exhibit relative good performance over other algorithms. Additionally, TT-TR-WP and TT-TR-CG are competitive in unconstrained problems, and the former has wide applicability while the latter has strong robustness. Moreover, the proposed algorithms are both more outstanding than the baseline algorithms in terms of applicability and robustness.

A large population island framework for the unconstrained binary quadratic problem

The unconstrained binary quadratic problem is an NP-hard problem and has applications in many fields. Recently, the problem has attracted much interest in the field of quantum optimization, as it is directly related to the Ising problem in physics and the development of quantum computers. However, effectively solving large instances of this problem remains a major challenge for existing solution methods. To advance the state of the art in solving the problem on a large scale, we propose an evolutionary algorithm with a very large population organized in different islands and integrating a new pairing and recombination method to produce promising offspring in each generation. Numerous experiments are conducted to evaluate the effects of different pairing strategies, crossovers, and migration topologies. This research has led to the discovery of new bounds for difficult instances of the maximum cut problem, which has been transformed using the binary quadratic formulation.

A novel unconstrained methodology for economic analysis of the level of repair with a case study of a multi-indenture and multi-echelon repair network

The multi-indenture and multi-echelon economic analysis of repair levels is commonly modelled as a constrained 0–1 programming problem. However, with the increase in the scale of the decision variables, the proportion of feasible solutions in the solution space sharply decreases, posing a challenge for traditional solution methods. In this paper, the transformation of the initial problem into an unconstrained integer programming problem is demonstrated. Specifically, the total number of feasible decisions can be solved analytically through a decision flow method, and a bijective relationship between decision sequence numbers and feasible decisions is established using a mixed-radix system, converting the optimization variables to feasible decision sequence numbers; in this way, the initial problem becomes unconstrained. The advantages of this approach include the following: (1) All the solutions in the transformed solution space are feasible, which significantly improves the efficiency of the solution and reduces the dimensionality of the decision variables; and (2) the proposed model is highly flexible in terms of considering additional variables, such as decision time, transportation modes, and repair locations. Furthermore, the sequence numbers of each feasible solution are transformed into binary sequences, and the binary particle swarm optimizer (BPSO) algorithm can be employed to solve the model. The proposed methodology is applied to the case of a three-indenture and three-echelon repair network considering multiple fault modes, and the results are compared with those of previous studies. The results validate the rationality and substantial advantages of the proposed methodology for economic analysis in terms of computational speed and convergence performance.

A new control method for leader–follower consensus problem of uncertain constrained nonlinear multi-agent systems

The design of Adaptive Fuzzy Sliding-Mode Control (AFSMC) is discussed in this study for the distributed leader–follower consensus problem of nonlinear uncertain multi-agent systems with input and state constraints. Avoiding the critical circumstances related to the violation of input and state constraints is an important issue in the design of consensus control systems. The limitations of existing adaptive consensus controllers developed for full-state constrained systems motivated the present novel method without restrictive structural assumptions. To convert the original problem to an unconstrained consensus control problem, the proposed control system uses a barrier function-based state transformation method and the input-state linearization methodology. As a result, each agent’s adaptive fuzzy sliding-mode control input, which consists of a fuzzy system and a robust term, is constructed based on the derived representation of the system dynamics. The fuzzy system is used to approximate the dynamic equations of the agent, its neighbors, and maybe the leader in each local feedback control law structure, and the robust term is designed to compensate for the fuzzy approximation error. The magnitudes of the robust terms and the output vectors of the fuzzy systems are also determined using the proposed adaptation laws. The second Lyapunov theorem and Barbalat’s lemma are used to verify the closed-loop system’s asymptotic stability. The effectuality of the proposed methodology is confirmed by numerical simulations for several Autonomous Underwater Vehicles (AUVs). Simulation results show that by means of the proposed AFSMC, the leader–follower consensus is achieved without state constraint violation and performance degradation.

Multi-objective reinforcement learning-based approach for pressurized water reactor optimization

A novel method, the Pareto Envelope Augmented with Reinforcement Learning (PEARL), has been developed to address the challenges posed by multi-objective problems, particularly in the field of engineering where the evaluation of candidate solutions can be time-consuming. PEARL distinguishes itself from traditional policy-based multi-objective Reinforcement Learning methods by learning a single policy, eliminating the need for multiple neural networks to independently solve simpler sub-problems. Several versions inspired from deep learning and evolutionary techniques have been crafted, catering to both unconstrained and constrained problem domains. Curriculum Learning (CL) is harnessed to effectively manage constraints in these versions. PEARL’s performance is first evaluated on classical multi-objective benchmarks. Additionally, it is tested on two practical PWR core Loading Pattern (LP) optimization problems to showcase its real-world applicability. The first problem involves optimizing the Cycle length (LC) and the rod-integrated peaking factor (FΔh) as the primary objectives, while the second problem incorporates the mean average enrichment as an additional objective. Furthermore, PEARL addresses three types of constraints related to boron concentration (Cb), peak pin burnup (Bumax), and peak pin power (Fq). The results are systematically compared against conventional approaches from stochastic optimization. Notably, PEARL, specifically the PEAL-NdS ( ▪ ) variant, efficiently uncovers a Pareto front without necessitating additional efforts from the algorithm designer, as opposed to a single optimization with scaled objectives. It also outperforms the classical approach across multiple performance metrics, including the Hyper-volume. Future works will encompass a sensitivity analysis of hyper-parameters with statistical analysis to optimize the application of PEARL and extend it to more intricate problems.

On the equivalence between Value-at-Risk- and Expected Shortfall-based risk measures in non-concave optimization

We study a non-concave optimization problem in which an insurance company maximizes the expected utility of the surplus under a risk-based regulatory constraint. The non-concavity does not stem from the utility function, but from non-linear functions related to the terminal wealth characterizing the surplus. For this problem, we consider four different prevalent risk constraints (Expected Shortfall, Expected Discounted Shortfall, Value-at-Risk, and Average Value-at-Risk), and investigate their effects on the optimal solution. Our main contributions are in obtaining an analytical solution under each of the four risk constraints in the form of the optimal terminal wealth. We show that the four risk constraints lead to the same optimal solution, which differs from previous conclusions obtained from the corresponding concave optimization problem under a risk constraint. Compared with the benchmark unconstrained utility maximization problem, all the four risk constraints effectively and equivalently reduce the set of zero terminal wealth, but do not fully eliminate this set, indicating the success and failure of the respective financial regulations.1

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.

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.

New approach to solve unconstrained binary quadratic problem

New approach to solve unconstrained binary quadratic problem

An explicit spectral Fletcher–Reeves conjugate gradient method for bi-criteria optimization

Abstract In this paper, we propose a spectral Fletcher–Reeves conjugate gradient-like method for solving unconstrained bi-criteria minimization problems without using any technique of scalarization. We suggest an explicit formulae for computing a descent direction common to both criteria. The latter further verifies a sufficient descent property that does not depend on the line search nor on any convexity assumption. After proving the existence of a bi-criteria Armijo-type stepsize, global convergence of the proposed algorithm is established. Finally, some numerical results and comparisons with other methods are reported.

Network traffic recovery from link-load measurements using tensor triple decomposition strategy for third-order traffic tensors

Network traffic data is the pivot of input in many network tasks but its direct measurement can be insufferably costly. In this paper, we propose a network traffic recovery method which only requires the conveniently measurable link-load traffics. We arrange the traffic data as a third-order tensor and utilize the triple decomposition technique proposed very recently by Qi et al. (2021). The studied model is a differentialble unconstrained minimization problem, which can be efficiently solved by a Barzilai–Borwein (BB) gradient algorithm. We prove that the generated iteration sequence can globally converge to a certain stationary point of the objective function. The numerical simulations on three open-source traffic datasets demonstrate the superiority of our method in comparison with other state-of-the-art algorithms.