Nonconvex Multiobjective Optimization Research Articles

Universal nonmonotone line search method for nonconvex multiobjective optimization problems with convex constraints

Universal nonmonotone line search method for nonconvex multiobjective optimization problems with convex constraints

An effective subgradient algorithm via Mifflin’s line search for nonsmooth nonconvex multiobjective optimization

We propose a descent subgradient algorithm for unconstrained nonsmooth nonconvex multiobjective optimization problems. To find a descent direction, we present an iterative process that efficiently approximates the ɛ-subdifferential of each objective function. To this end, we develop a new variant of Mifflin’s line search in which the subgradients are arbitrary and its finite convergence is proved under a semismooth assumption. To reduce the number of subgradient evaluations, we employ a backtracking line search that identifies the objectives requiring an improvement in the current approximation of the ɛ-subdifferential. Meanwhile, for the remaining objectives, new subgradients are not computed. Unlike bundle-type methods, the proposed approach can handle nonconvexity without the need for algorithmic adjustments. Moreover, the quadratic subproblems have a simple structure, and hence the method is easy to implement. We analyze the global convergence of the proposed method and prove that any accumulation point of the generated sequence satisfies a necessary Pareto optimality condition. Furthermore, our convergence analysis addresses a theoretical challenge in a recently developed subgradient method. Through numerical experiments, we observe the practical capability of the proposed method and evaluate its efficiency when applied to a diverse range of nonsmooth test problems.

A dynamic multiple proximal bundle algorithm for locally Lipschitz multiobjective optimization problems

We propose a method for nonsmooth nonconvex multiobjective optimization problems that is based on a bundle-type approach. The proposed method employs an extension of the redistributed proximal bundle algorithm and uses an augmented improvement function to handle different objectives. In which, at each iteration, a common piecewise linear model is used to approximate the augmented improvement function. Contrary to many existing multiobjective optimization methods, this algorithm works directly with objective functions, without using any kind of a priori chosen parameters or employing any scalarization. Under appropriate assumptions, we discuss convergence to points which satisfy a necessary condition for Pareto optimality. We provide numerical results for a set of nonsmooth convex and nonconvex multiobjective optimization problems in the form of tables, figures and performance profiles. The numerical results confirm the superiority of the proposed algorithm in the considered test problems compared to other multiobjective solvers, in the computational effort necessary to compute weakly Pareto stationary points.

Convergence and complexity guarantees for a wide class of descent algorithms in nonconvex multi-objective optimization

We address conditions for global convergence and worst-case complexity bounds of descent algorithms in nonconvex multi-objective optimization. Specifically, we define the concept of steepest-descent-related directions. We consider iterative algorithms taking steps along such directions, selecting the stepsize according to a standard Armijo-type rule. We prove that methods fitting this framework automatically enjoy global convergence properties. Moreover, we show that a slightly stricter property, satisfied by most known algorithms, guarantees the same complexity bound of O(ϵ−2) as the steepest descent method.

Characterizing the nonemptiness and compactness of weakly efficient solution sets for non-convex multiobjective optimization problems

In this paper, we demonstrate that non-convex multiobjective optimization problems have nonempty and compact weakly efficient solution sets if and only if finite scalar optimization problems have nonempty and compact solution sets under mild conditions, which partially reduces the gap between non-convex and convex multiobjective problems in characterizing the nonemptiness and compactness of weakly efficient solution sets. Two examples are provided to support the findings.

Convexificators for nonconvex multiobjective optimization problems with uncertain data: robust optimality and duality

In this paper, we investigate robust optimality conditions and duality for a class of nonconvex multiobjective optimization problems with uncertain data in the worst case by the upper semi-regular convexificator. The Fermat principle for a locally Lipschitz function is presented in terms of the upper semi-regular convexificator. We establish robust necessary optimality conditions of the Fritz-John type and KKT type for the uncertain nonconvex multiobjective optimization problems. In addition, robust sufficient optimality conditions as well as saddle point conditions are derived under the generalized ∂ ˆ ∗ -pseudoquasiconvexity and generalized convexity, respectively. The robust duality relations between the original problem and its mixed robust dual problem are obtained under a generalized pseudoconvexity assumption.

Combined heat and power economic emission dispatch using dynamic switched crowding based multi-objective symbiotic organism search algorithm

Combined heat and power economic emission dispatch (CHPEED) problem is a highly complex, non-linear, non-convex multi-objective optimization problem due to two conflicting objectives and various operational constraints such as valve-point loading effect, power transmission loss, prohibited operating zone, and the feasible operating region of combined heat and power unit. In order to overcome these challenges, it is necessary to design an algorithm that exhibits a search behavior, which is suitable for the characteristics of objective and constraint space of the CHPEED problem. For these reasons, a dynamic switched crowding based multi-objective symbiotic organism search (DSC-MOSOS) algorithm was designed to meet the requirements and geometric space of the CHPEED problem. By applying the DSC method in the MOSOS algorithm, it was aimed to improve the exploration ability, to strengthen exploitation-exploration balance, and to prevent the catching into local solution traps. A comprehensive experimental study was carried out to prove the performance of the proposed algorithm on IEEE CEC 2020 multi-modal multi-objective problems (MMOPs) and CHPEED problem. In the experimental study conducted among eleven versions of MOSOS variations created with DSC-method and the base MOSOS algorithm on IEEE CEC 2020 MMOPs, according to Friedman scores based on the four performance metrics, the base MOSOS algorithm ranked the last. In other experimental study, the best DSC-MOSOS variant was applied to solve the CHPEED problem, where 5-, 7-, 10- and 14-unit test systems and eight case studies were considered. The important points of this study were that 10-unit and 14-unit test systems were presented to the literature, and the prohibited operating zone was considered in CHPEED problem for the first time. According to the results obtained from eight case studies obtained from the DSC-MOSOS and fourteen competitor algorithms, while the improvement in cost was between 0.2% and 16.55%, the reduction of the emission value was between 0.2 kg and 42.97 kg compared to the competitor algorithms. On the other hand, the stability of the DSC-MOSOS and the base MOSOS was evaluated using stability analysis. While the MOSOS algorithms was not able to perform a success in any case study, the DSC-MOSOS was achieved an average success rate with 91.16%. Thus, the performance of the DSC-MOSOS over the MOSOS was verified by the results of experimental studies and analysis.

A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization

This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are covered and form the motivation of our studies. The presented algorithm is based on a branch-and-bound method in the pre-image space, a technique which was already successfully applied for continuous nonconvex multiobjective optimization. However, extending this method to the mixed-integer setting is not straightforward, in particular with regard to convergence results. More precisely, new branching rules and lower bounding procedures are needed to obtain an algorithm that is practically applicable and convergent for multiobjective mixed-integer optimization problems. Corresponding results are a main contribution of this paper. What is more, for improving the performance of this new branch-and-bound method we enhance it with two types of cuts in the image space which are based on ideas from multiobjective mixed-integer convex optimization. Those combine continuous convex relaxations with adaptive cuts for the convex hull of the mixed-integer image set, derived from supporting hyperplanes to the relaxed sets. Based on the above ingredients, the paper provides a new multiobjective mixed-integer solver for convex problems with a stopping criterion purely in the image space. What is more, for the first time a solver for multiobjective mixed-integer nonconvex optimization is presented. We provide the results of numerical tests for the new algorithm. Where possible, we compare it with existing procedures.

Self-calibration algorithm for installation angle deviation of bionic polarization compound eyes.

A self-calibration algorithm based on unsupervised optimization for polarizer installation angle deviation is proposed and used in a multi-aperture bionic polarization compound eye system. To simplify calibration operation, under the condition that the calibration-polarized light information is unknown, this algorithm fully exploits redundancy and random polarization information in the scene, and uses a non-convex multi-objective discrete parameter sorting optimization method to achieve angle self-calibration. Compared with ordinary calibration procedures, the algorithm requires less stringent conditions, achieves online calibration and is more accurate. It also can be applied to camera polarization arrays, division-of-focal-plane polarization cameras, and other polarization devices.

Image regularity conditions for nonconvex multiobjective optimization problems with applications

The main purpose of this paper is to investigate Lagrange saddle points and calmness properties for a nonconvex multiobjective optimization problem by virtue of image regularity conditions. Following along with the image space analysis, the quasi-interior, quasi-relative interior and linear image regularity conditions are presented. Simultaneously, some equivalent characterizations to Fritz John and Karush/Kuhn-Tucker saddle points are established respectively by virtue of the proposed image regularity conditions. Moreover, some calmness properties are also obtained by means of local conic image regularity conditions in the image space of the multiobjective optimization problem.

PSACCF: Prioritized Online Slice Admission Control Considering Fairness in 5G/B5G Networks

5G/B5G is intended to support various services with the assistance of network slices, and each slice requires adequate resources to provide the prenegotiated service quality. The Slice Admission Control (SAC) algorithm is a necessity for Slice Providers (SPs) to guarantee the Quality of Service (QoS) and Quality of Experience (QoE) of each admitted request. In such circumstances, concerns regarding the priority of services and the fairness of resource allocation are meaningful topics. However, these two issues do not receive sufficient research attention simultaneously in the literature. Here, we study the SAC problem in 5G/B5G networks, aiming to enhance the fairness degree while satisfying the necessary priority requirements. We first reinterpret priority as a higher Cumulative Service Acceptance Ratio (CSAR) and adopt the uniformity of adjacent CSAR gaps to reflect fairness. The SAC problem is formulated as a nonlinear and nonconvex multiobjective optimization problem. Thus, we propose a heuristic algorithm called Prioritized Slice Admission Control Considering Fairness (PSACCF) to solve it. This approach introduces the resource efficiency of services to amend priority violations and then promotes fairness by setting the target CSARs for each service type and improving their actual CSARs. Numerous simulations are conducted to compare the performance of PSACCF with that of Moderate High Priority First (MHPF) and Aggressive High Priority First (AHPF). The results show that PSACCF can achieve at least a 33.6% improvement in fairness degree and a higher minimum average resource utilization while maintaining a nearly identical priority indicator to that of the compared methods.

A class of infeasible proximal bundle methods for nonsmooth nonconvex multi-objective optimization problems

A class of infeasible proximal bundle methods for nonsmooth nonconvex multi-objective optimization problems

A multi-objective natural aggregation algorithm for optimizing user allocation matrix in visible light communication

Allocating users to LED panels in a visible light communication (VLC) environment is a non-convex multi-objective constrained optimization problem. The computational complexity and accuracy of the optimization are the key factors to deal with in a VLC environment. In this manuscript, initially the user allocation problem is formulated as a single objective optimization problem by using sum-utility function of data rate and blockage probability. Then these two contradictory objective functions are simultaneously optimized by a new multi-objective natural aggregation algorithm (MONAA). This proposed algorithm is inspired by the aggregation behavior of cockroaches which help them to determine the darkest spot in a search space. The performance of this multi-objective algorithm is compared with NSGA-II, MOPSO, MOEA/D and MOSOS on eight unconstrained and six constrained mathematical function optimizations. In single objective and multi-objective cases, superior performance is demonstrated by the natural aggregation algorithm for solving the user allocation matrix optimization.

A Trust Region Reduced Basis Pascoletti-Serafini Algorithm for Multi-Objective PDE-Constrained Parameter Optimization

In the present paper non-convex multi-objective parameter optimization problems are considered which are governed by elliptic parametrized partial differential equations (PDEs). To solve these problems numerically the Pascoletti-Serafini scalarization is applied and the obtained scalar optimization problems are solved by an augmented Lagrangian method. However, due to the PDE constraints, the numerical solution is very expensive so that a model reduction is utilized by using the reduced basis (RB) method. The quality of the RB approximation is ensured by a trust-region strategy which does not require any offline procedure, in which the RB functions are computed in a greedy algorithm. Moreover, convergence of the proposed method is guaranteed and different techniques to prevent the excessive growth of the number of basis functions are explored. Numerical examples illustrate the efficiency of the proposed solution technique.

Spectral-Energy Efficiency Trade-Off Based Design for Hybrid TDMA-NOMA System

The combination of time division multiple access (TDMA) and non-orthogonal multiple access (NOMA), referred to as hybrid TDMA-NOMA system, is considered as a potential solution to meet the unprecedented requirements for future wireless networks. While recent resource allocation techniques aiming to individually maximize either spectral efficiency (SE) or energy efficiency (EE), this paper considers an SE-EE trade-off based technique for a hybrid TDMA-NOMA system. This design offers an additional degree of freedom in resource allocation. The proposed design is formulated as a non-convex multi-objective optimization (MOO) problem. The MOO framework is reformulated as a single-objective optimization (SOO) problem by combining the multi-objectives through a weighted-sum objective function. With this, each of the original objectives is assigned with a weight factor to reflect its importance in the design. Then, the sequential convex approximation (SCA) and a second-order cone (SOC) approach are jointly utilized to deal with the non-convexity issues of the SOO problem. Simulation results reveal that the proposed trade-off based design strikes a good balance between the objective functions, while meeting the instantaneous requirements of the system.

An improved proximal method with quasi-distance for nonconvex multiobjective optimization problem

Abstract Multiobjective optimization is the optimization with several conflicting objective functions. However, it is generally tough to find an optimal solution that satisfies all objectives from a mathematical frame of reference. The main objective of this article is to present an improved proximal method involving quasi-distance for constrained multiobjective optimization problems under the locally Lipschitz condition of the cost function. An instigation to study the proximal method with quasi distances is due to its widespread applications of the quasi distances in computer theory. To study the convergence result, Fritz John’s necessary optimality condition for weak Pareto solution is used. The suitable conditions to guarantee that the cluster points of the generated sequences are Pareto–Clarke critical points are provided.

Performance Space Supported Design of Analog Electronic Circuits

Performance spaces as representations for attainable image spaces of analog electronic circuits are well-suited for supporting their conventional design process. The design task can be formulated as non-convex multi-objective optimization problem. We apply a numerical approach based on the normal-boundary intersection method to model two example circuits. Furthermore, the impact of a power limitation and relaxation of design constraints on the resulting performance spaces is evaluated. Thus, we show how these models are supporting conclusions for important design decisions. We also present a concept for visualizing active constraints on the performance space boundary to directly identify the responsible circuit parts.

Motion Pattern Optimization and Energy Analysis for Underwater Glider Based on the Multi-Objective Artificial Bee Colony Method

Underwater gliders are prevailing in oceanic observation nowadays for their flexible deployment and low cost. However, the limited onboard energy constrains their application, hence the motion pattern optimization and energy analysis are the key to maximizing the range of the glider while maintaining the acceptable navigation preciseness of the glider. In this work, a Multi-Objective Artificial Bee Colony (MOABC) algorithm is used to solve the constrained hybrid non-convex multi-objective optimization problem about range and accuracy of gliders in combination with specific glider dynamics models. The motion parameters Pareto front that balances the navigational index referring to range and preciseness are obtained, relevant gliding profile motion results are simulated simultaneously, and the results are compared with the conventional gliding patterns to examine the quality of the solution. Comparison shows that, with the utilization of the algorithm, glider voyage performance with respect to endurance and preciseness can be effectively improved.

Convergence of a nonmonotone projected gradient method for nonconvex multiobjective optimization

Convergence of a nonmonotone projected gradient method for nonconvex multiobjective optimization

Perturbation of Image and conjugate duality for vector optimization

<p style='text-indent:20px;'>This paper aims at employing the image space approach to investigate the conjugate duality theory for general constrained vector optimization problems. We introduce the concepts of conjugate map and subdifferential by using two types of maximums. We also construct the conjugate duality problems via a perturbation method. Moreover, the separation condition is proposed by means of vector weak separation functions. Then, it is proved to be a new sufficient condition, which ensures the strong duality theorem. This separation condition is different from the classical regular conditions in the literature. Simultaneously, the application to a nonconvex multi-objective optimization problem is shown to verify our main results.</p>