Unconstrained Multi-objective Optimization Problems Research Articles

Pareto-critical equilibrium condition for non-linear multiobjective optimization problems with applications

The new (necessary) Pareto-critical equilibrium condition (PCEC) discussed in this study is based on the Pareto-critical condition for non-linear optimization problems. The proposed condition is evaluated by defining a multiobjective optimization problems (MOOPs) subject to a set of constraints. Then this MOOPs is converted to a single objective optimization problem by using the available optimization techniques, for instance, the weighted sum method, and solved by the non-linear Nelder-Mead simplex method. The obtained solution points satisfy the proposed PCEC of an unconstrained multiobjective optimization problem (MOOPs) under the convexity assumption. This PCEC is evaluated in a search that either results in a point inside the feasible region or a point for which the PCEC holds. The resulting Pareto-critical equilibrium condition is globally valid for convex problems. Numerical experiments are carried out in order to clarify the proposed condition's validity. Moreover, the proposed PCEC is used to place a downlink transmit beamforming system base station in an ideal position as a first application and is also employed in situations where there are multiple agents working towards a common goal as a second application.

Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-Like Systems

We derive efficient algorithms to compute weakly Pareto optimal solutions for smooth, convex and unconstrained multiobjective optimization problems in general Hilbert spaces. To this end, we define a novel inertial gradient-like dynamical system in the multiobjective setting, which trajectories converge weakly to Pareto optimal solutions. Discretization of this system yields an inertial multiobjective algorithm which generates sequences that converge weakly to Pareto optimal solutions. We employ Nesterov acceleration to define an algorithm with an improved convergence rate compared to the plain multiobjective steepest descent method (Algorithm 1). A further improvement in terms of efficiency is achieved by avoiding the solution of a quadratic subproblem to compute a common step direction for all objective functions, which is usually required in first-order methods. Using a different discretization of our inertial gradient-like dynamical system, we obtain an accelerated multiobjective gradient method that does not require the solution of a subproblem in each step (Algorithm 2). While this algorithm does not converge in general, it yields good results on test problems while being faster than standard steepest descent.

Multiobjective-Based Constraint-Handling Technique for Evolutionary Constrained Multiobjective Optimization: A New Perspective

Multiobjective-based constraint-handling techniques are popular in evolutionary constrained single-objective optimization. However, most of these techniques run into troubles when dealing with constrained multiobjective optimization problems (CMOPs). That is, they have difficulty optimizing too many objective functions, are ineffective in maintaining population diversity, or are challenged in establishing appropriate additional objective functions. As a remedy to these limitations, we propose a novel technique called NRC for handling CMOPs. The novelty of NRC lies in its three sorting procedures: nondominated sorting, reversed nondominated sorting, and constrained crowding distance sorting, which are performed in sequence to provide driving forces toward the Pareto front (PF) of a transformed unconstrained multiobjective optimization problem (treating the overall constraint violation as an additional objective function), the boundary front, and the constrained PF, respectively. With the combination of these three different forces, NRC can conveniently approach the desired PF from diverse search directions. The effectiveness of NRC is experimentally verified. Also, we incorporate NRC into a two-archive mechanism and develop a novel constrained multiobjective evolutionary algorithm, called NRC2. Comprehensive experiments on 49 benchmark CMOPs and 21 real-world ones demonstrate that NRC2 is significantly superior or comparable to six state-of-the-art constrained evolutionary multiobjective optimizers on most test instances.

A limited memory Quasi-Newton approach for multi-objective optimization

In this paper, we deal with the class of unconstrained multi-objective optimization problems. In this setting we introduce, for the first time in the literature, a Limited Memory Quasi-Newton type method, which is well suited especially in large scale scenarios. The proposed algorithm approximates, through a suitable positive definite matrix, the convex combination of the Hessian matrices of the objectives; the update formula for the approximation matrix can be seen as an extension of the one used in the popular L-BFGS method for scalar optimization. Equipped with a Wolfe type line search, the considered method is proved to be well defined even in the nonconvex case. Furthermore, for twice continuously differentiable strongly convex problems, we state global and R-linear convergence to Pareto optimality of the sequence of generated points. The performance of the new algorithm is empirically assessed by a thorough computational comparison with state-of-the-art Newton and Quasi-Newton approaches from the multi-objective optimization literature. The results of the experiments highlight that the proposed approach is generally efficient and effective, outperforming the competitors in most settings. Moreover, the use of the limited memory method results to be beneficial within a global optimization framework for Pareto front approximation.

Mobile Robot Path Planning with a Novel Multiobjective Slime Mold Algorithm

In the path planning problem of mobile robot in static complex obstacle environment, the original slime mold algorithm (SMA) has some shortcomings, such as initial solution error and easy to fall into local optimum, which leads to low accuracy of robot path planning. A novel Baldwin learning effect mobile robot Path planning with Multi‐Objective Slime Mold Algorithm (BPMOSMA) is proposed to solve the related problems. The Baldwin learning effect is introduced into to guide the development direction of slime mold in the phase of slime mold surrounding food and the k‐nearest neighbor measure is added to prune nondominated solutions to generate a well‐distributed solution set. For comparison purposes, five state‐of‐the‐art multiobjective optimization algorithms are carried on nine multiobjective test problems with kinds of types of Pareto set. The qualitative and quantitative experimental results show that BPMOSMA is very competitive performance on unconstrained multi‐objective optimization problems. BPMOSMA is finally successfully applied in the mobile robot path planning problem.

Variable Metric Method for Unconstrained Multiobjective Optimization Problems

In this paper, we propose a variable metric method for unconstrained multiobjective optimization problems (MOPs). First, a sequence of points is generated using different positive definite matrices in the generic framework. It is proved that accumulation points of the sequence are Pareto critical points. Then, without convexity assumption, strong convergence is established for the proposed method. Moreover, we use a common matrix to approximate the Hessian matrices of all objective functions, along which a new nonmonotone line search technique is proposed to achieve a local superlinear convergence rate. Finally, several numerical results demonstrate the effectiveness of the proposed method.

A novel multi-level population hybrid search evolution algorithm for constrained multi-objective optimization problems

Constrained multi-objective optimization problem (CMOP) considers the convergence, diversity and feasibility of the population in the optimization process, so it is challenging to find desirable solutions of CMOP. Existing evolutionary multi-objective optimization algorithms have good performance on unconstrained multi-objective optimization problems, but have difficulties in solving CMOPs in discrete feasible regions. Aiming at this issue, this paper proposes a novel multi-level population hybrid search evolution algorithm (MLHSEA). First, a new multi-level hybrid search strategy (i.e. MHSS) is designed in the algorithm, which divides the population into three-level subpopulations based on Pareto ranks, constraint violation degree values, and feasible thresholds. Each subpopulation has its own unique evolution strategy to maximize the evolutionary potential of each subpopulation, which is beneficial to make some feasible solutions break through the discrete feasible region and reach the Pareto frontier. Then, a new population fusion degree strategy (i.e. PFDS) is proposed to timely perform population fusion and information exchange according to the population fusion degree (PFD) of each sub-population, thus improving the searchability of the target space. Finally, a novel detection reset strategy (i.e. DRS) is proposed for the lowest inferior subpopulation. This strategy can make inferior subpopulations avoid unnecessary evolutionary iterations and improve population diversity. Based on constrained test suites with four different characteristics, the experimental results show that the proposed MLHSEA outperforms other state-of-the-art constrained multi-objective optimization algorithms in performance.

Nonmonotone trust region algorithm for solving the unconstrained multiobjective optimization problems

Nonmonotone trust region algorithm for solving the unconstrained multiobjective optimization problems

Correction to: manuscript entitled: MO-NFSA for solving unconstrained multi-objective optimization problems

Correction to: manuscript entitled: MO-NFSA for solving unconstrained multi-objective optimization problems

MO-NFSA for solving unconstrained multi-objective optimization problems

Normative Fish Swarm Algorithm (NFSA) is a novel variant of the Artificial Fish Swarm Algorithm (AFSA) proposed in 2019, and has been proven effective in solving single-objective optimization problems. Inspired by the potential of NFSA, this paper proposes an evolutionary multi-objective optimization algorithm, namely “Multi-Objective Normative Fish Swarm Algorithm (MO-NFSA)”. However, due to the fact that NFSA was originally modeled for single-objective optimization, a certain degree of transformation, modification, and additional strategies (related to multi-objective optimization features) must be integrated into the original NFSA as part of MO-NFSA modeling for further extension. This article adopts a total of 15 multi-objective optimization test cases in any category of fixed-dimensional, non-fixed-dimensional (ZDT set) or scalable multi-objective (DTLZ set) optimization types. These multi-objective optimization test cases are used to compare the performance of MO-NFSA with other comparative algorithms. However, performance evaluation of multi-objective optimization is a daunting task. Their performance results can only be digitized through quality indicators (i.e., performance metrics), which are mainly tested from three different aspects of high-quality approximation: convergence, uniformity, and spread. Here, quality indicators including generation distance (GD), spacing (S), and spread-delta (∆/∆*) are used to check the quality performance of each algorithm based on the corresponding approximation. In the work, 20 simulations are run, and hence, 20 sets of data are collected. The collected results prove that MO-NFSA is superior to other comparison algorithms in all aspects of high-quality approximation. MO-NFSA can solve different types of multi-objective optimization problems.

Inverse multiobjective optimization: Inferring decision criteria from data

It is a challenging task to identify the objectives on which a certain decision was based, in particular if several, potentially conflicting criteria are equally important and a continuous set of optimal compromise decisions exists. This task can be understood as the inverse problem of multiobjective optimization, where the goal is to find the objective function vector of a given Pareto set. To this end, we present a method to construct the objective function vector of an unconstrained multiobjective optimization problem (MOP) such that the Pareto critical set contains a given set of data points with prescribed KKT multipliers. If such an MOP can not be found, then the method instead produces an MOP whose Pareto critical set is at least close to the data points. The key idea is to consider the objective function vector in the multiobjective KKT conditions as variable and then search for the objectives that minimize the Euclidean norm of the resulting system of equations. By expressing the objectives in a finite-dimensional basis, we transform this problem into a homogeneous, linear system of equations that can be solved efficiently. Potential applications of this approach include the identification of objectives (both from clean and noisy data) and the construction of surrogate models for expensive MOPs.

Accelerated Diagonal Steepest Descent Method for Unconstrained Multiobjective Optimization

In this paper, we propose two methods for solving unconstrained multiobjective optimization problems. First, we present a diagonal steepest descent method, in which, at each iteration, a common diagonal matrix is used to approximate the Hessian of every objective function. This method works directly with the objective functions, without using any kind of a priori chosen parameters. It is proved that accumulation points of the sequence generated by the method are Pareto-critical points under standard assumptions. Based on this approach and on the Nesterov step strategy, an improved version of the method is proposed and its convergence rate is analyzed. Finally, computational experiments are presented in order to analyze the performance of the proposed methods.

A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems

Multi-objective optimization problems (MOPs) naturally arise in many applications. Since for such problems one can expect an entire set of optimal solutions, a common task in set based multi-objective optimization is to compute N solutions along the Pareto set/front of a given MOP. In this work, we propose and discuss the set based Newton methods for the performance indicators Generational Distance (GD), Inverted Generational Distance (IGD), and the averaged Hausdorff distance Δp for reference set problems for unconstrained MOPs. The methods hence directly utilize the set based scalarization problems that are induced by these indicators and manipulate all N candidate solutions in each iteration. We demonstrate the applicability of the methods on several benchmark problems, and also show how the reference set approach can be used in a bootstrap manner to compute Pareto front approximations in certain cases.

State-transition simulated annealing algorithm for constrained and unconstrained multi-objective optimization problems

In this article, a novel multi-objective optimization algorithm based on a state-transition simulated annealing algorithm (MOSTASA) is proposed, in which four state-transition operators for generating candidate solutions and the Pareto optimal solution is obtained by combining it with the concept of Pareto dominance and then storing it in a Pareto archive. To ensure the uniform distribution of the Pareto optimal solution, we define a crowded comparison operator to update the Pareto archive. Simulation experiments were conducted on several standard constrained and unconstrained multi-objective problems, in which convergence and spacing metrics were used to assess the performance of the MOSTASA. The test results manifest that the MOSTASA can converge to the true Pareto-optimal front, and the solution distribution is uniform. Compared to the performance of other multi-objective optimization algorithms, the proposed algorithm is more efficient and reliable.

On q-Quasi-Newton’s Method for Unconstrained Multiobjective Optimization Problems

A parameter-free optimization technique is applied in Quasi-Newton’s method for solving unconstrained multiobjective optimization problems. The components of the Hessian matrix are constructed using q-derivative, which is positive definite at every iteration. The step-length is computed by an Armijo-like rule which is responsible to escape the point from local minimum to global minimum at every iteration due to q-derivative. Further, the rate of convergence is proved as a superlinear in a local neighborhood of a minimum point based on q-derivative. Finally, the numerical experiments show better performance.

A superlinearly convergent nonmonotone quasi-Newton method for unconstrained multiobjective optimization

ABSTRACT We propose and analyse a nonmonotone quasi-Newton algorithm for unconstrained strongly convex multiobjective optimization. In our method, we allow for the decrease of a convex combination of recent function values. We establish the global convergence and local superlinear rate of convergence under reasonable assumptions. We implement our scheme in the context of BFGS quasi-Newton method for solving unconstrained multiobjective optimization problems. Our numerical results show that the nonmonotone quasi-Newton algorithm uses fewer function evaluations than the monotone quasi-Newton algorithm.

A multi-agent complex network algorithm for multi-objective optimization

To deal with the multi-objective optimization problems (MOPs), this study proposes a new Multi-Objective Multi-Agent Complex Network Optimization Algorithm called MOMCNA based on the idea of Cellular genetic algorithm (CGA) and the Multi-agent complex network. Compared with the traditional CGA for multi objective problem, the individuals in the population of MOMCNA have more features of intelligent agent, the new form of neighborhood for the population, private archive for individuals, the new strategy of “local-global” genetic operator and the chaotic mutation are proposed in the new algorithm to balance the convergence and diversity of the algorithm. Seventeen unconstrained multi-objective optimization problems and seven many-objective problems are introduced and tested to evaluate the new algorithm, in addition, the classical traffic assignment problem based on different system optimum principle is also established to evaluate the new algorithm. The comparison between MOMCNA and other classical algorithms shows that the proposed MOMCNA proves to be competitive in dealing with multi-objective and many-objective optimization problems and the structure of the complex network made up of population also has effect on algorithm’s performance.

On q-Newton’s method for unconstrained multiobjective optimization problems

In this paper, we present a method of so-called q-Newton’s type descent direction for solving unconstrained multiobjective optimization problems. The algorithm presented in this paper is implemented by applying an independent parameter q (quantum) in an Armijo-like rule to compute the step length which guarantees that the value of the objective function decreases at every iteration. The search processes gradually shift from global in the beginning to local as the algorithm converges due to q-gradient. The algorithm is experimented on 41 benchmark/test functions which are unimodal and multi-modal with 1, 2, 3, 4, 5, 10 and 50 different dimensions. The performance of the proposed method is confirmed by comparing with three existing schemes.

Combined weighted multi-objective optimizer for instance reduction in two-class imbalanced data problem

Instance reduction from class-balanced data has been investigated in much research. However, there is a lack of studies on class-imbalanced data. Learning from imbalanced data lately has attracted a lot of attention due to the practical applications. In the case of two-class imbalanced data, the instances from one class, majority class, are more numerous than the instances from the other class, which is a minority class. The present paper aims to introduce a new instance reduction method that preserves between-class distributions in the balanced data and handles minority class instance reduction in two-class imbalanced data, efficiently. The proposed method solves the instance reduction issue from an unconstrained multi-objective optimization problem aspect. Accordingly, a new combined weighted optimizer is designed. By employing the chaotic krill herd evolutionary algorithm, both the minority and majority class spaces with the accelerated convergence are explored. Through this method, the original data set is purged of those instances that decrease accuracy, and Gmean. The performance has been evaluated on both imbalanced and balanced data sets collected from the UCI repository by the 10-fold cross-validation method. Evaluations show that the proposed method outperforms state-of-the-art methods in terms of classification accuracy, Gmean, reduction rates, and computational time.

On <i>q</i>-steepest descent method for unconstrained multiobjective optimization problems

The <i>q</i>-gradient is the generalization of the gradient based on the <i>q</i>-derivative. The <i>q</i>-version of the steepest descent method for unconstrained multiobjective optimization problems is constructed and recovered to the classical one as <i>q</i> equals 1. In this method, the search process moves step by step from global at the beginning to particularly neighborhood at last. This method does not depend upon a starting point. The proposed algorithm for finding critical points is verified in the numerical examples.