Classical Nonlinear Optimization Research Articles

Optimization Models for Urban Traffic Management

The main control tool for traffic management in urban areas is traffic light settings. The goal is to decrease the queue lengths at intersections. Usually, the duration of the green light of the traffic light is used for control. The control approach is based on the so-called “store-and-forward” model. However, this model does not reflect the stochastic nature of traffic dynamics. This study presents a model with some probabilistic conditions approximating real traffic behavior. An additional contribution concerns the definition of a bi-level optimization model that simultaneously optimizes the green light and traffic light cycle duration of an urban network of four intersections. Three traffic management optimization problems are defined and solved. Their solutions are graphically illustrated and commented on. Bi-level optimization outperforms by giving lower values of queue lengths compared to classical and stochastic nonlinear optimization problems in the considered network.

The effectiveness of the sports management system in Europe: High achievements, public funding and a healthy lifestyle

The article proposes an original approach to evaluating the effectiveness of the sports management system, which differs from the classical linear and non-linear optimization methods due to its use of quantile regression models. Three main indicators were chosen for creating the quantile regressions: (i) number of participants in the Olympic Games - as a feature of the effectiveness of the high-achievement sports; (ii) total number of people engaged in sports - as a feature of an effective mass sports system, which ensures a healthy lifestyle of the population; and (iii) state expenses on recreational and sports services. The quality of quantile regressions was validated by the Fisher test and the two-factor variance analysis. The research focused on the data of 30 European countries, which officially and publicly provide access to statistical reports on these issues. The countries were divided into two groups with above-average and below-average effectiveness of the sports management system. Two hypotheses were proposed and tested in the study. Hypothesis H1 was that the effectiveness of high-achievement sports depends on the amount of state funding for developing the sports industry. Research results confirmed this hypothesis for countries with an above-average level of sports management system effectiveness, but refuted it for the other group. Hypothesis H2 stated that the more massive the development of sport in a country, i.e., the more people are engaged in sports and lead a healthy lifestyle, the more likely this country is to achieve victories in high-achievement sports. This hypothesis was confirmed for both groups of countries.

Magnetic particle tracking: A semi-algebraic solution

Magnetic Particle Tracking (MPT) is a relatively new non-invasive measurement technique which is often used to study dense granular flow. Its basic principle relies on tracking the movement of a single magnetic tracer by means of measuring the magnetic field strength at a suitable distance from the tracer. By assumption of a magnetic dipole and the use of minimization techniques, both location and orientation of the tracer can be determined. MPT is therefore uniquely suited for the study of non-spherical particles. The performance of the localization is largely dependent on the signal-to-noise ratio and very often relies on nonlinear optimization techniques, as the definition of the magnetic field generated by a dipole is highly nonlinear and has five degrees of freedom. In this paper, we present a semi-algebraic solution by decoupling the estimation of the position and orientation in separate algebraic solutions. The two estimates are mutually dependent, necessitating an iterative approach between the two. The main benefits of this new approach is in the speed and robustness of the algorithm, which are much higher than for the classical constrained nonlinear optimization techniques.

Discrete Dynamical System Approaches for Boolean Polynomial Optimization

In this article, we discuss the numerical solution of Boolean polynomial programs by algorithms borrowing from numerical methods for differential equations, namely the Houbolt scheme, the Lie scheme, and a Runge-Kutta scheme. We first introduce a quartic penalty functional (of Ginzburg-Landau type) to approximate the Boolean program by a continuous one and prove some convergence results as the penalty parameter $$\varepsilon $$ converges to 0. We prove also that, under reasonable assumptions, the distance between local minimizers of the penalized problem and the set $$\{\pm 1\}^n$$ is of order $$O(\sqrt{n}\varepsilon )$$ . Next, we introduce algorithms for the numerical solution of the penalized problem, these algorithms relying on the Houbolt, Lie and Runge-Kutta schemes, classical methods for the numerical solution of ordinary or partial differential equations. We performed numerical experiments to investigate the impact of various parameters on the convergence of the algorithms. We have tested our ODE approaches and compared with the classical nonlinear optimization solver IPOPT and a quadratic binary formulation approach (QB-G) as well as an exhaustive method using parallel computing techniques. The numerical results on various datasets (including small and large-scale randomly generated synthetic datasets of general Boolean polynomial optimization problems, and a large-scale heterogeneous MQLib benchmark dataset of Max-Cut and Quadratic Unconstrained Binary Optimization (QUBO) problems) show good performances for our ODE approaches. As a result, our ODE algorithms often converge faster than the other compared methods to better integer solutions of the Boolean program.

Novel Real Valued Improved Coral-Reef Optimization Algorithm for Optimal Integration of Classified Distributed Generators

In recent times, Distributed Generation (DG) penetration, especially, through Renewable Energy Sources (RES) has been growing immensely due to marginal carbon footprints. Furthermore, they make system more reliable by minimizing the voltage deviation of the distribution network. Nevertheless, to exploit the DGs in the best manner, they need to be sited and rated optimally. Diverse solutions exist for the problem mainly classified as: analytical approaches, classical non-linear optimization algorithms and meta-heuristic methods. With the objective of minimizing power loss, this manuscript proposes a novel hybrid meta-heuristics approach: Particle Swarm Optimization-Coral Reef Optimization (PSO-CRO) for identifying an optimum positioning and rating of Type-1, Type-2 & Type-3 DGs (at 0.82 optimal power factor) in IEEE 33, 69 & 118 bus Radial Distributed System (RDS). Furthermore, the results from the proposed hypothesis are compared with its prevalent peers, namely, PSO, CRO, Gravitational Search Algorithm (GSA), PSO-GSA and PSO-Grey Wolf Optimization (PSO-GWO) etc. The results of the proposed algorithm are also compared with the results of GAMS/CONOPT commercial solver. The simulation results prove the robustness, higher efficacy and faster convergence of the proposed method when applied to larger distribution systems.

Neural Tensor Network Training Using Meta-Heuristic Algorithms for RDF Knowledge Bases Completion

ABSTRACTNeural tensor network (NTN) has been recently introduced to complete Resource Description Framework (RDF) knowledge bases, which has been the state-of-the-art in the field so far. An RDF knowledge base includes some facts from the real world shown as RDF “triples.” In the previous methods, an objective function has been used for training this type of network, and the network parameters should have been set in a way to minimize the function. For this purpose, a classic nonlinear optimization method has been used. Since many replications are needed in this method to get the minimum amount of the function, in this paper, we suggest to combine meta-heuristic optimization methods to minimize the replications and increase the speed of training consequently. So, this problem will be improved using some meta-heuristic algorithms in this new approach to specify which algorithm will get the best results on NTN and its results will be compared with the results of the former methods finally.

Performance of hybrid quantum-classical variational heuristics for combinatorial optimization.

The recent literature on near-term applications for quantum computers contains several examples of the applications of hybrid quantum-classical variational approaches. This methodology can be applied to a variety of optimization problems, but its practical performance is not well studied yet. This paper moves some steps in the direction of characterizing the practical performance of the methodology, in the context of finding solutions to classical combinatorial optimization problems. Our study is based on numerical results obtained applying several classical nonlinear optimization algorithms to Hamiltonians for six combinatorial optimization problems; the experiments are conducted via noise-free classical simulation of the quantum circuits implemented in Qiskit. We empirically verify that: (1) finding the ground state is harder for Hamiltonians with many Pauli terms; (2) classical global optimization methods are more successful than local methods due to their ability of avoiding the numerous local optima; (3) there does not seem to be a clear advantage in introducing entanglement in the variational form.

APPLICATION AND PERFORMANCE ANALYSIS OF A NEW BUNDLE ADJUSTMENT MODEL

Abstract. As the basis for photogrammetry, Bundle Adjustment (BA) can restore the pose of cameras accurately, reconstruct the 3D models of environment, and serve as the criterion of digital production. For the classical nonlinear optimization of BA model based on the Euclidean coordinate, it suffers the problem of being seriously dependent on the initial values, making it unable to converge fast or converge to a global minimum. This paper first introduces a new BA model based on parallax angle feature parametrization, and then analyses the applications and performance of the model used in photogrammetry field. To discuss the impact and the performance of the model (especially in aerial photogrammetry), experiments using two aerial datasets under different initial values were conducted. The experiment results are better than some well-known software packages of BA, and the simulation results illustrate the stability of the new model than the normal BA under the Euclidean coordinate. In all, the new BA model shows promising applications in faster and more efficient aerial photogrammetry with good convergence and fast convergence speed.

Real-time smart distribution system reconfiguration using complementarity

Smart Radial Distribution Systems (SRDS) of the future will have improved reliability, performance and flexibility in operation by using algorithms such as optimal reconfiguration in real-time in their distribution management systems. However, today, optimal reconfiguration algorithms are largely academic because of challenges such as they (1) depend upon heuristic techniques that require repeated runs and are not suitable for real-time applications, (2) do not guarantee an optimal solution and, (3) do not provide insight into solution space.In order to realize SRDS of the future, a real-time optimal reconfiguration algorithm is proposed, which uses a classic nonlinear optimization technique and guarantees an optimal solution in the least time. The method is based upon a complementarity technique that transforms discontinuous solution spaces into continuous, enabling use of classical nonlinear optimization techniques without resorting to heuristics.Using the complementarity technique, a nonlinear optimization formulation and classical solution method is needed to optimally reconfigure a SRDS and to minimize losses while obtaining an acceptable voltage solution is proposed.This is successfully demonstrated on 7-bus, 33-bus, and 69-bus distribution systems and the results are compared with those available in literature with respect to solution time, accuracy in results and robustness of the proposed algorithm and demonstrate superiority of the proposed technique.

A Comparison of the Embedding Method With Multiparametric Programming, Mixed-Integer Programming, Gradient-Descent, and Hybrid Minimum Principle-Based Methods

In recent years, the embedding approach for solving switched optimal control problems has been developed in a series of papers. However, the embedding approach, which advantageously converts the hybrid optimal control problem to a classical nonlinear optimization, has not been extensively compared with alternative solution approaches. The goal of this paper is thus to compare the embedding approach with multiparametric programming, mixed-integer programming [mixed integer programming (MIP), commercial (CPLEX)], and gradient-descent-based methods in the context of five recently published examples: 1) a spring-mass system; 2) moving-target tracking for a mobile robot; 3) two-tank filling; dc-dc boost converter; and 5) skid-steered vehicle. A sixth example, an autonomous switched 11-region linear system, is used to compare a hybrid minimum principle method and traditional numerical programming. For a given performance index (PI) for each case, cost and solution times are presented. It is shown that there are numerical advantages of the embedding approach: lower PI cost (except in some instances when autonomous switches are present), generally faster solution time, and convergence to a solution when other methods may fail. In addition, the embedding method requires no ad hoc assumptions (e.g., predetermined mode sequences) or specialized control models. Theoretical advantages of the embedding approach over the other methods are also described; guaranteed existence of a solution under mild conditions, convexity of the embedded hybrid optimization problem (under the customary conditions on the PI), solvability with traditional techniques (e.g., sequential quadratic programming) avoiding the combinatorial complexity in the number of modes/discrete variables of MIP, applicability to affine nonlinear systems, and no need to explicitly assign discrete/mode variables to autonomous switches. Finally, common misconceptions regarding the embedding approach are addressed, including whether it uses an average value control model (no), whether it is necessary to tweak the algorithm to obtain bang-bang solutions (no), whether it requires infinite switching to implement embedded solution (no), and whether it has real-time capability (yes).

The Application of the Linear Adaptive Genetic Algorithm to Optimal Power Flow Problem

This paper proposes a Linear Adaptive Genetic Algorithm (LAGA) for optimal power flow (OPF) problem. The proposed approach offers faster convergence than the standard genetic algorithm. In this study, LAGA is applied to the 6-bus system and the IEEE 14-bus power system. Shunt capacitance, transformer taps and generator voltages are used as control system variables to minimize the system power loss. The output of the systems under investigation is compared with the output of a classical nonlinear optimization routine to evaluate the impact of LAGA technique to OPF. Moreover, to validate the performance of LAGA approach, a demonstration and comparison with earlier published results is presented. Simulation results are found to be effective and promising.

Steiner Path for Trees

The Steiner minimum tree (SMT) problem is one of the classic nonlinear combinatorial optimization problems for centuries. The Steiner tree problem finds the minimum cost tree connecting a given set of nodes called required nodes in a given undirected weighted graph. In this minimum cost Steiner tree due to the reduction of path-cost, some nonrequired nodes are also used while finding the SMT, which are called Steiner nodes. The Steiner path problem comes from the Steiner tree problem of finding the minimum cost path connecting a given set of nodes called required nodes in a given undirected weighted graph. The Steiner path problem can be characterized as either a decision or optimization problem. This paper focuses on solving the Steiner path as a decision problem. The Steiner Path decision problem finds whether there is a path connecting a given set of nodes called required nodes in a given binary tree. The decision that whether there exists a Steiner path or not can help in further solving the Steiner path optimization problem. Since the Steiner tree decision problem has wide scales, we should use an efficient algorithm with least complexity. In this paper we have defined the steiner path decision problem, characterized sufficient conditions for finding the steiner path and presented an algorithm for finding the existence of a steiner path in a 2-tree.

Geometry-Experiment Algorithm for Steiner Minimal Tree Problem

It is well known that the Steiner minimal tree problem is one of the classical nonlinear combinatorial optimization problems. A visualization experiment approach succeeds in generating Steiner points automatically and showing the system shortest path, named Steiner minimum tree, physically and intuitively. However, it is difficult to form stabilized system shortest path when the number of given points is increased and irregularly distributed. Two algorithms, geometry algorithm and geometry-experiment algorithm (GEA), are constructed to solve system shortest path using the property of Delaunay diagram and basic philosophy of Geo-Steiner algorithm and matching up with the visualization experiment approach (VEA) when the given points increase. The approximate optimizing results are received by GEA and VEA for two examples. The validity of GEA was proved by solving practical problems in engineering, experiment, and comparative analysis. And the global shortest path can be obtained by GEA successfully with several actual calculations.

Identification of unknown groundwater pollution sources using classical optimization with linked simulation

Identification of unknown groundwater pollution sources still remains a challenging problem. Some of the complexities include: sparse observation data, substantial variation in magnitude of the source fluxes distributed over space and time, uncertainties in the imposed initial and boundary conditions. Methodologies already developed for optimal identification of pollution sources using concentration measurements and hydraulic data suffer from a number of limitations. As an alternative, a source identification methodology is proposed that uses a classical nonlinear optimization model linked to a flow and transport simulation model. The groundwater flow and transport simulator is linked to the nonlinear optimization model as an external module. The essential link between the simulator and the optimization method are the derivatives or gradient information required for the optimization algorithm. This proposed methodology is potentially applicable to large scale study areas and does not posses some of the computational limitations of some earlier developed methodologies, using nonlinear programming with the flow and transport process governing equations embedded as equality constraints within the optimization model. Performance of the proposed source identification methodology using spatiotemporal pollutant concentration measurements are evaluated by solving illustrative problems. Two different optimization formulations models are developed. The relative importance of the model formulations is demonstrated in terms of computational efficiency. The limited performance evaluations reported here demonstrate the potential applicability of the developed methodology using nonlinear programming and linked flow and transport simulation model for a fairly large study area with multiple unknown pollution sources.

3D Points Registration Algorithm with Engineering Constraints

Constraints of certain key point are often required in the estimation of the best position and orientation for a rigid body in cases of engineering application.Constraints are seldom involved in former point registration method,which result in deficiency of method for constraint rigid body adjustment.A registration method with engineering constraint such as two points’ symmetry is proposed,as well as points on certain plane and points on certain line.The constraints are added into registration objective function for optimization by using multi-objective model.Three rotation parameters and three translation parameters are obtained by using classical Newton nonlinear optimization method.Standard examples are presented for the demonstration of the algorithm.

Confidence structural robust design and optimization under stiffness and load uncertainties

Most of the structural behavior constraints involved in structural robust design and optimization are non-convex in nature. Therefore if local optimality criteria based optimization algorithms are employed to find the worst case structural responses that are used for examining the feasibility of a given design, it is highly possible that the optimization process will get stuck in a local optimum. If this is the case, the reliability of a “robust” design cannot be guaranteed, at least theoretically. The aim of the present paper is to develop some new formulations and the corresponding numerical algorithms to find the confidence optimal design under non-probabilistic stiffness and load uncertainties. To this end, two Bi-level program formulations for confidence robust design are proposed. In order to ensure the strict feasibility of the optimal solution, in the lower-level of program, the constraints are imposed on the confidence upper bounds of the structural responses, which can be obtained efficiently by solving some convex linear semi-definite programs (LSDPs). Based on the sensitivity analysis of the lower-level LSDP problem, the upper level programs are then solved by employing the classical gradient-based nonlinear optimization algorithms. Furthermore, for the case of stiffness uncertainty, a single-level nonlinear semi-definite programming (NSDP) formulation is also proposed and its mathematical properties are analyzed. Numerical examples show that confidence robust optimal design can be obtained via the proposed approaches effectively without resorting to too many computational efforts.

Relação entre modelos de programação não-linear com incerteza no conjunto de restrições

Este trabalho demonstra de forma analítica e numérica a relação entre dois métodos, Trappey et al. (1988) e Xu (1989), da literatura que resolvem problemas de programação não-linear com incertezas no conjunto de restrições. Uma análise comparativa entre o desempenho para a obtenção da solução ótima dos métodos de otimização não-linear clássicos e dos métodos de otimização não-linear nebulosos, apresentados também neste trabalho. Para tal comparação, serão apresentados dois problemas que foram modelados em termos de programação não-linear clássico, os quais permitem a introdução de incertezas nas restrições. Com base na análise dos problemas propostos em Xu (1989), verificou-se que os dois métodos descritos fornecem resultados similares, conforme algumas condições.

Adaptive Parametric Identification Scheme for a Class of Nondeteriorating and Deteriorating Nonlinear Hysteretic Behavior

The adaptive parametric identification of deteriorating and nondeteriorating nonlinear hysteretic phenomena is considered using a generalization of Masing model based on the observed memory behavior of distributed element models. The model permits a parametric identification to be performed using nonlinear optimization techniques for arbitrary response time histories. A changing objective function, defined as the normalized force estimation error over a shifting window of recent data, is employed so that classic nonlinear optimization techniques can be used for the adaptive identification problem. A variation of the steepest descent method is used with significant modifications. To achieve the best performance for any given problem, a set of a priori numeric tests are suggested to design the identification scheme. The design identification scheme exhibits a very good performance in identifying the correct values of the parameters and is rather robust in dealing with noise. The proposed approach has applications to adaptive identification of much wider types of nonlinear rate-dependent hysteretic behavior. Also, the set of guidelines proposed by the authors is a contribution toward having more effective autonomous identification schemes, using minimal information about the model and input.

A robust online parametric identification method for non-deteriorating and deteriorating distributed element models with viscous damping

The online parametric identification of deteriorating and non-deteriorating distributed element models (DEMs) with viscous damping is studied using a generalization of Masing model to provide the proper framework for identification. The approach renders the hysteretic response of the DEM into a time-independent single-valued mapping from equivalent displacement values into equivalent force values, while considering the effect of damping as a parallel element. This approach allows for parametric identification of this non-linear rate-dependent hysteretic behavior to be performed using non-linear optimization techniques. A changing objective function, defined as a norm of force estimation error over a shifting window of recent data, is employed so that classic non-linear optimization techniques can be used for the online identification problem. A variation of the steepest descent method is used with significant modifications. Special measures are taken to guarantee robustness of the results in presence of noise. The results show that the proposed identification method exhibits a very good performance in identifying the correct values of the parameters in real time, and is robust in dealing with noise. The proposed method can be applied to many other types of hysteretic behavior as well.

Optimization of activated sludge designs using genetic algorithms

We describe a framework in which a genetic algorithm (GA) and a static activated sludge (AS) treatment plant design model (WRC AS model) are used to identify low cost activated sludge designs that meet specified effluent limits (e.g. for BOD, N, and P). Once the user has chosen a particular process (Bardenpho, Biodenipho, UCT or SBR), this approach allows the parameterizations for each AS unit process to be optimized systematically and simultaneously. The approach is demonstrated for a wastewater treatment plant design problem and the GA-based performance is compared to that of a classical nonlinear optimization approach. The use of GAs for multiobjective problems such as AS design is demonstrated and their application for reliability-based design and alternative generation is discussed.