Solution Of Optimization Problem Research Articles

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.

The method of determining optimal control of the thermoelastic state of piece-homogeneous body using a stationary temperature field

This paper proposes a new highly effective method for determining the optimal control of the stress-strain state of spatially multi-connected composite bodies using a stationary temperature field. The proposed method is considered based on the example of a stationary axisymmetric thermoelastic problem for a space with a spherical inclusion and cavity. The proposed method is based on the generalized Fourier method and reduces the original problem to an equivalent problem of optimal control, in which the state of the object is determined by an infinite system of linear algebraic equations, the right-hand side of which parametrically depends on the control. At the same time, the functional of the cost of the initial problem is transformed into a quadratic functional, which depends on the state of the equivalent system and parametrically on the control. The limitation on the temperature distribution is replaced by the value of the control norm in the space of square summable sequences. In fact, this paper considers for the first time the problem of optimal control of an infinite system of linear algebraic equations and develops a method for its solution. The proposed method is based on presenting the solutions of infinite systems in a parametric form, which makes it possible to reduce equivalent problem to the problem of conditional extremum of a quadratic functional, which explicitly depends on the control. A further solution to this problem A further solution to this problem is found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. The method developed in this paper is strictly justified. For all infinite systems, the Fredholm property of their operators is proved. As an important result necessary for substantiation, for the first time, an estimate from below of the module of the multi-parameter determinant of the resolving system of the boundary value problem of conjugation – space with a spherical inclusion – was obtained when solving it using the Fourier method. The theorem that establishes the conditions for the existence and uniqueness of the solution of equivalent problem or optimal control problem without restrictions in the space of square summable sequences is proved. The numerical algorithm is based on a reduction method for infinite systems of linear algebraic equations. Estimates of the practical accuracy of the numerical algorithm demonstrated the stability of the method and sufficiently high accuracy even with close location of the boundary surfaces. Graphs showing the optimal temperature distribution for various geometric parameters of the problem and their analysis are provided. The proposed method extends to boundary value problems with different geometries.

Low-Carbon Optimal Configuration of Integrated Electricity and Natural Gas Energy System with Life-Cycle Carbon Emission

In response to the challenges of global warming and the development of A low-carbon economy, the integrated electricity and natural gas energy system (IEGES) is known as an important structure for future energy supply; thus, its planning and design must take low-carbon and environmental protection factors into account. Regarding carbon emissions as an optimization criterion, this paper built life-cycle carbon emission models of IEGES components. Then, taking the capacities of the energy resources, storage and conversion units of IEGES as the optimization variables, a multi-objective optimization configuration model was established considering the annual investment operation cost and the life-cycle carbon emissions. The multi-objective model was transformed into a single-objective one by an ε-constraint approach and the polynomial fitting method was employed to obtain the value of ε for obtaining uniformly distributed Pareto sets. Based on the fuzzy entropy weight method and the fuzzy affiliation degree approach, the obtained Pareto sets were ranked and the solution with the highest ranking value was selected as the optimal solution for the original problem. Finally, the configuration schemes were analyzed from the perspectives of economy, carbon emission and renewable energy utilization, and the effectiveness and rationality of the proposed optimization method were verified through MATLAB simulation.

An Algorithm for Solving Three-dimensional Assignment Problem

This article presents a algorithm for solving Three-dimensional assignment problem. Firstly, decompose the three-dimensional cubic matrix corresponding to the three-dimensional assignment problem into multiple two-dimensional planar matrices, and obtain that the assignment problems corresponding to these two-dimensional planar matrices have the same feasible solution as the original three-dimensional assignment problem. Then, the leading principal submatrix algorithm is used to solve each two-dimensional assignment problem. The characteristic of the leading principal submatrix algorithm is that each operation only needs to consider the local (leading principal submatrix) of the assignment matrix of the two-dimensional assignment matrix, without considering the entire assignment matrix. Starting from the first-order leading principal submatrix of the assignment matrix, Through the same solution transformation, the row minimum element of the leading principal submatrix of each order of the assignment matrix is found step by step, and the optimal solution of the two-dimensional assignment problems are obtained. Finally, by comparing the optimal solutions of these two-dimensional assignment problems, the optimal solutions of the original three-dimensional assignment problems are obtained. This algorithm can find the optimal solution for three-dimensional assignment problems in a patterned manner, facilitating computer programming and handling assignment problems with a large number of people and tasks.

Multi-objective RIME algorithm-based techno economic analysis for security constraints load dispatch and power flow including uncertainties model of hybrid power systems

In recent times, the landscape of power systems has undergone significant evolution, particularly with the integration of diverse renewable energy sources (RESs). This advancement presents an invaluable opportunity to enhance energy efficiency in the modern power grid, primarily by bolstering the role of stochastic RESs. The challenge lies in the optimal power flow (OPF), a multifaceted and non-linear optimization challenge that grows more complex with the inclusion of stochastic RESs that aims to optimize the allocation of power system resources to minimize the operational cost while maintaining the stability and security of the system. Addressing this, the current study introduces an innovative optimization approach, the Multi-Objective RIME (MORIME) algorithm. Drawing inspiration from the physical phenomenon of rime-ice, called the RIME, the MORIME seeks to effectively tackle OPF issues. This algorithm enhances solution accuracy by smartly dividing with non-dominated sorting and crowding distance mechanism. The proposed OPF model incorporates three types of RESs: solar photovoltaic, wind and small-scale hydropower units. While uncertainties in wind speed and solar irradiation are managed through Monte Carlo simulations, the small hydro unit is considered a constant power source. The efficacy of the MORIME algorithm is tested on IEEE 30 bus systems and results indicate that the MORIME method identifies the optimal solution for the multi-objective OPF problem while satisfying the power system constraints, thereby proving its effectiveness and superiority over MOWOA, MOGWO, MOALO, MOMRFO and MOAGDE in terms of Hyper Volume (HV) and reciprocal of Pareto Sets Proximity (1/PSP) metrices. The MORIME source code is available at: https://github.com/kanak02/MORIME

System Engineering based on Remarkable Geometric Properties of Space

In this paper, we regard designing systems based on remarkable geometric properties of space, namely valuable rotational symmetry and asymmetry harmonious, using schematic and diagrammatic presentations of the systems. Moreover, the relationships are a way to comprehend original information to serve as a source of research and designing the systems. The objective of the future methodology is the advanced study of spatial geometric harmony as profiting information for expansion fundamental and applied researches for optimal solutions of technological problems in systems engineering. These systems engineering designs make it possible to improve the quality indices of devices or systems concerning performance reliability, code immunity, and the other operating indices of the systems. As examples, both up to 25% errors of lengths correcting code and high-speed self-error-correcting vector data code formed under a toroidal coordinate system are presented.

Methodology for Selecting Optimal Routes for the Transportation of Dangerous Goods in Conditions of Risk Uncertainty

Abstract * A risk situation is a type of uncertainty situation where events are likely to occur. In other words, risk is an estimated probability, and uncertainty is something that cannot be quantified. The greater the uncertainty when making a decision, the greater the degree of risk. The investigation is devoted to the development of a unified methodology for selecting optimal (from the proposed alternatives) routes for transporting dangerous goods (any of the 9 standard classes) by road. A model for determining two groups of risk components for transporting dangerous goods on separate routes is proposed, namely: parameters of risks of probability of occurrence of emergency situations and parameters that affect the risks of complexity of eliminating their possible consequences. This opened up opportunities for correct selection of criteria when forming a multi-criteria optimization problem of transportation. Mathematical algorithms for a comprehensive solution of the multi-criteria optimization problem of routing dangerous goods transportation are developed using the classical criteria of Laplace, Wald, Savage, and Hurwitz. To confirm the legal capacity of the proposed approach and mathematical apparatus, the optimal route for transportation of cylinders with technical gases through the transport network of Lutsk was calculated. Possible alternative transportation routes have been identified and safety criteria have been formalized for two groups of risk components

The Multiobjective Control Based on Tolerance Optimization in a Multienergy System

To address the issue of multiobjective control in multienergy systems with diverse operational objectives, a two-stage optimization framework based on expected point tolerance has been proposed in this paper. In the first stage, a single objective function is used for optimization control to obtain the expected point of the multiobjective optimization problem. Then, in the second stage, by defining the allowable deviation between each optimization objective and the expected point, the original multiobjective optimization problem is transformed into a single objective optimization problem solution with tolerance measurement. Finally, in the simulation scene of a multienergy system, it is demonstrated that compared with the optimal results under each single objective method, the proposed method increases power line loss, maximum voltage deviation, new energy consumption, and economy by 2.22, 2.30, 1.02, and 2.45 times, respectively. Compared with the suboptimal results, the proposed method reduces power line loss by 22.26, 1.74, 1.09, and 0.97 times, respectively. Combining the shape of the Pareto frontier, it is demonstrated that the proposed method can comprehensively consider the needs of multiple power optimization objectives for forming a more reasonable and effective system optimization scheduling and also provide a new approach for solving multiobjective optimization problems.

Higher-order KKT optimality conditions through contingent derivatives for constrained nonsmooth vector equilibrium problems

In this paper, we deal with some higher-order optimality conditions for local strict efficient solutions to a nonsmooth vector equilibrium problem with set, cone and equality constraints. For this aim, the concept of m−stable and m−steady functions (m≥2 and integer) for single-valued functions and some constraint qualifications of higher order in terms of contingent derivatives are proposed accordingly. We analyze the sum calculus rule of mth-order adjacent set, mth-order interior set, asymptotic mth-order tangent cone and asymptotic mth-order adjacent cone. Subsequently, we employ the obtained calculus rules to treat KKT necessary and sufficient optimality conditions of higher order in terms of contingent derivatives for the mth-order (local) strict efficient solutions to such problem. Simultaneously, we employ these rules to study KKT higher-order optimality conditions for such efficient solutions for a nonsmooth vector optimization problem with constraints. Some illustrative examples are provided to demonstrate the main results of the literature.

Taking the human out of decomposition-based optimization via artificial intelligence, Part II: Learning to initialize

The repeated solution of large-scale optimization problems arises frequently in process systems engineering tasks. Decomposition-based solution methods have been widely used to reduce the corresponding computational time, yet their implementation has multiple steps that are difficult to configure. We propose a machine learning approach to learn the optimal initialization of such algorithms which minimizes the computational time. Active and supervised learning is used to learn a surrogate model that predicts the computational performance for a given initialization. We apply this approach to the initialization of Generalized Benders Decomposition for the solution of mixed-integer model predictive control problems. The surrogate models are used to find the optimal initialization, which corresponds to the number of initial cuts that should be added in the master problem. The results show that the proposed approach can lead to a significant reduction in solution time, and active learning can reduce the data required for learning.

Bearing Fault Diagnosis via Stepwise Sparse Regularization with an Adaptive Sparse Dictionary.

Vibration monitoring is one of the most effective approaches for bearing fault diagnosis. Within this category of techniques, sparsity constraint-based regularization has received considerable attention for its capability to accurately extract repetitive transients from noisy vibration signals. The optimal solution of a sparse regularization problem is determined by the regularization term and the data fitting term in the cost function according to their weights, so a tradeoff between sparsity and data fidelity has to be made inevitably, which restricts conventional regularization methods from maintaining strong sparsity-promoting capability and high fitting accuracy at the same time. To address the limitation, a stepwise sparse regularization (SSR) method with an adaptive sparse dictionary is proposed. In this method, the bearing fault diagnosis is modeled as a multi-parameter optimization problem, including time indexes of the sparse dictionary and sparse coefficients. Firstly, sparsity-enhanced optimization is conducted by amplifying the regularization parameter, making the time indexes and the number of atoms adaptively converge to the moments when impulses occur and the number of impulses, respectively. Then, fidelity-enhanced optimization is carried out by removing the regularization term, thereby obtaining the high-precision reconstruction amplitudes. Simulations and experiments verify that the reconstruction accuracy of the SSR method outperforms other sparse regularization methods under most noise conditions, and thus the proposed method can provide more accurate results for bearing fault diagnosis.

Optimal solution of nonlinear 2D variable-order fractional optimal control problems using generalized Bessel polynomials

This study aims to propose a new optimization method based on the generalized Bessel polynomials (GBPs) as a class of basis functions for a category of nonlinear two-dimensional variable-order fractional optimal control problems (N-2D-VOFOCPs) involved in fractional-order dynamical systems and Caputo derivatives. For the optimal solution of such problems, the optimization method is developed on the basis of operational matrices (OMs) scheme of derivatives, 2D Gauss–Legendre quadrature rule, and Lagrange multiplier technique. The state and control functions are expanded in terms of the GBPs to reduce the complexity of these problems. The proposed method focuses on a system of nonlinear algebraic equations in the process of finding solution to the problems. The convergence of the method based on GBPs is proved, and the accuracy of the method is analyzed by solving several examples.

General spatial photonic Ising machine based on the interaction matrix eigendecomposition method

The spatial photonic Ising machine has achieved remarkable advancements in solving combinatorial optimization problems. However, it still remains a huge challenge to flexibly map an arbitrary problem to the Ising model. In this paper, we propose a general spatial photonic Ising machine based on the interaction matrix eigendecomposition method. The arbitrary interaction matrix can be configured in the two-dimensional Fourier transformation based spatial photonic Ising model by using values generated by matrix eigendecomposition. The error in the structural representation of the Hamiltonian decreases substantially with the growing number of eigenvalues utilized to form the Ising machine. In combination with the optimization algorithm, as low as ∼65% of the eigenvalues are required by intensity modulation to guarantee the best probability of optimal solution for a 20-vertex graph Max-cut problem, and this percentage decreases to below ∼20% for near-zero probability. The 4-spin experiments and error analysis demonstrate the Hamiltonian linear mapping and ergodic optimization. Our work provides a viable approach for spatial photonic Ising machines to solve arbitrary combinatorial optimization problems with the help of the multi-dimensional optical property.

A cooperative neural dynamic model for solving general convex nonlinear optimization problems with fuzzy parameters and an application in manufacturing systems

SummaryIn the presented study, the solution of the fuzzy nonlinear optimization problems (FNLOPs) is calculated using a recurrent neural network (RNN) model. Since there is a few research for solving FNLOP by RNN's, we give a new approach to solve the problem. By reducing the original program to an interval problem and then weighting problem, the Karush–Kuhn–Tucker (KKT) conditions are given. Moreover, we use the KKT conditions into a RNN as an important tool to solve the problem. Besides, the global convergence properties and the Lyapunov stability of the dynamic model are studied in this study. In the final step, some illustrative examples are considered to establish the obtained results. Reported results are compared with some others network models.

Computationally efficient solution of mixed integer model predictive control problems via machine learning aided Benders Decomposition

Mixed integer Model Predictive Control (MPC) problems arise in the operation of systems where discrete and continuous decisions must be taken simultaneously to compensate for disturbances. The efficient solution of mixed integer MPC problems requires the computationally efficient online solution of mixed integer optimization problems, which are generally difficult to solve. In this paper, we propose a machine learning-based branch and check Generalized Benders Decomposition algorithm for the efficient solution of such problems. We use machine learning to approximate the effect of the complicating variables on the subproblem by approximating the Benders cuts without solving the subproblem, therefore, alleviating the need to solve the subproblem multiple times. The proposed approach is applied to a mixed integer economic MPC case study on the operation of chemical processes. We show that the proposed algorithm always finds feasible solutions to the optimization problem, given that the mixed integer MPC problem is feasible, and leads to a significant reduction in solution time (up to 97% or 50×) while incurring small error (in the order of 1%) compared to the application of standard and accelerated Generalized Benders Decomposition.

WOA: Wombat Optimization Algorithm for Solving Supply Chain Optimization Problems

Supply Chain (SC) Optimization is a key activity in today’s industry with the goal of increasing operational efficiency, reducing costs, and improving customer satisfaction. Traditional optimization methods often struggle to effectively use resources while handling complex and dynamic Supply chain networks. This paper introduces a novel biomimetic metaheuristic algorithm called the Wombat Optimization Algorithm (WOA) for supply chain optimization. This algorithm replicates the natural behaviors observed in wombats living in the wild, particularly focusing on their foraging tactics and evasive maneuvers towards predators. The theory of WOA is described and then mathematically modeled in two phases: (i) exploration based on the simulation of wombat movements during foraging and trying to find food and (ii) exploitation based on simulating wombat movements when diving towards nearby tunnels to defend against its predators. The effectiveness of WOA in addressing optimization challenges is assessed by handling the CEC 2017 test suite across various problem dimensions, including 10, 30, 50, and 100. The findings of the optimization indicate that WOA demonstrates a strong ability to effectively manage exploration and exploitation, and maintains a balance between them throughout the search phase to deliver optimal solutions for optimization problems. A total of twelve well-known metaheuristic algorithms are called upon to test their performance against WOA in the optimization process. The outcomes of the simulations reveal that WOA outperforms the other algorithms, achieving superior results across most benchmark functions and securing the top ranking as the most efficient optimizer. Using a Wilcoxon rank sum test statistical analysis, it has been proven that WOA outperforms other algorithms significantly. WOA is put to the test with twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems to showcase its ability to solve real-world optimization problems. The results of the simulations demonstrate that WOA excels in real-world applications by delivering superior solutions and outperforming its competitors.

Optimization of Problems with Multi-Objective Functions and their Applications in Engineering

Many real-world optimization issues typically have multiple competing goals. There is generally no solution in those multi-objective optimization problems that optimizes all objective functions at the same time. Rather, "efficient" in terms of all objective function’s solutions known as Pareto optimum solutions are presented. We typically have a large number of Pareto-optimal options. As a result, we must choose a final solution from among Pareto optimal solutions while considering the objective function balance; this process is known as "trade-off analysis." It is not hyperbole to state that trade-off analysis is the most crucial task in multi-objective optimization. As a result, the methodology's ease of use and comprehension for trade-off analysis should be highlighted. The set of Pareto optimal solutions in the objective function space, or Pareto frontier, can be represented somewhat simply in circumstances when there are two or three objective functions. We can fully understand the trade-off relationship between objectives when we see Pareto boundaries. Thus, in scenarios with two or three objectives, it would be the most appropriate technique to represent Pareto borders. Reading the trade-off relationship between objectives with three dimensions, however, may be challenging. Nonetheless, Pareto frontier cannot be represented in situations where there are more than three objectives. In this case, interactive techniques can assist us in performing a local trade-off analysis that reveals a "certain" Pareto optimal option. Many techniques have been devised, with variations in which the Pareto optimal solution is displayed. The main concerns of such multi-objective optimization techniques, especially as they relate to engineering design problems, are covered in this study.

Unbalanced penalization: a new approach to encode inequality constraints of combinatorial problems for quantum optimization algorithms

Solving combinatorial optimization problems of the kind that can be codified by quadratic unconstrained binary optimization (QUBO) is a promising application of quantum computation. Some problems of this class suitable for practical applications such as the traveling salesman problem (TSP), the bin packing problem (BPP), or the knapsack problem (KP) have inequality constraints that require a particular cost function encoding. The common approach is the use of slack variables to represent the inequality constraints in the cost function. However, the use of slack variables considerably increases the number of qubits and operations required to solve these problems using quantum devices. In this work, we present an alternative method that does not require extra slack variables and consists of using an unbalanced penalization function to represent the inequality constraints in the QUBO. This function is characterized by larger penalization when the inequality constraint is not achieved than when it is. We evaluate our approach on the TSP, BPP, and KP, successfully encoding the optimal solution of the original optimization problem near the ground state cost Hamiltonian. Additionally, we employ D-Wave Advantage and D-Wave hybrid solvers to solve the BPP, surpassing the performance of the slack variables approach by achieving solutions for up to 29 items, whereas the slack variables approach only handles up to 11 items. This new approach can be used to solve combinatorial problems with inequality constraints with a reduced number of resources compared to the slack variables approach using quantum annealing or variational quantum algorithms.

On $$\varepsilon $$-quasi efficient solutions for fractional infinite multiobjective optimization problems with locally Lipschitz data

On $$\varepsilon $$-quasi efficient solutions for fractional infinite multiobjective optimization problems with locally Lipschitz data

HPB3C-3PG algorithm: A new hybrid global optimization algorithm and its application to plant classification

This paper proposes a hybrid bio-inspired search and optimization algorithm that combines the strengths of the PB3C (Parallel Big Bang Big Crunch) and 3PGA (3 Parent Genetic Algorithm) algorithms. The hybrid algorithm employs a single population-based evolutionary search coupled with multi-population parallel processing techniques to address optimization problems. The proposed algorithm is implemented in MATLAB software. We evaluate the performance of the proposed algorithm on the CEC2021 standard test bench suite. The performance of the proposed approach is compared with that of the other nine algorithms. The comparative analysis shows that the proposed hybrid PB3C and 3PGA algorithms performed better than the other nine optimization algorithms. Furthermore, this chapter proposes an HPB3C-3PGA-based approach to evolve the near-optimal architecture of CNN. The proposed plant image classification approach is implemented in Python and compared with 12 other approaches. The proposed approach achieved an accuracy of 98.96% on the Mendeley dataset and 98.97% on the CVIP100 dataset. The proposed approach outperforms all other approaches for the plant leaf classification problem. This research significantly contributes to overcoming limitations in existing approaches, providing a robust solution for optimization problems and image classification tasks.