Non-convex Multi-objective Optimization Problem Research Articles

Optimal Allocation of Flood Control Capacity for Multi-Reservoir Systems Using Multi-Objective Optimization Approach

Operation of multi-reservoir systems during flood periods is of great importance in the field of water resources management. This paper proposes a multi-objective optimization model with new formulation for optimal operation of multi-reservoir systems. In this model, the release rate and the flood control capacity of each reservoir is considered as decision variable and the resulting nonlinear non-convex multi-objective optimization problem is solved with e-constraint method through the mixed integer linear programming (MILP). Objective functions of the model are minimizing the flood damage at downstream sites and the loss of hydropower generation. The developed model is used to determine optimal operating strategies for Karkheh multi-reservoir system in southwestern Iran. For this purpose, the model is executed in two scenarios based on “two-reservoir” and “six-reservoir” systems and for floods with return periods of 25 and 50 years. The results show that in two-reservoir system, flood damage is at least about 114 million dollars and cannot be mitigated any further no matter how hydropower generation is managed. But, in the case of developing all six reservoirs, optimal strategies of coordinated operation can mitigate and even fully prevent flood damage.

A New Scalarization Technique and New Algorithms to Generate Pareto Fronts

We propose a new scalarization technique for nonconvex multiobjective optimization problems and establish its theoretical properties. By combining our new scalarization approach with existing grid ...

A Radial Boundary Intersection aided interior point method for multi-objective optimization

We propose a novel multi-objective optimization technique combining non-convex Radial Boundary Intersection based decomposition with an Interior Point method (which utilizes both line search and trust region steps) suitable for non-convex nonlinear optimization. Radial Boundary Intersection decomposes the multi-objective optimization problem into subproblems which are concerned with finding the solutions closest to a reference point along equally spaced lines emanating radially outwards from the latter point. The proposed approach is found to be able to generate good approximations of the Pareto front (including the periphery) by generating a sufficiently diverse set of Pareto optimal solutions. The proposed method is extensively tested on a large number of recent benchmark problems and real world problems and the performance is found to be favorable in comparison to those of some of the cutting-edge stochastic/evolutionary optimization algorithms that are commonly used to solve non-convex multi-objective optimization problems.

Generalized asymptotic functions in nonconvex multiobjective optimization problems

In this paper, we use generalized asymptotic functions and second-order asymptotic cones to develop a general existence result for the nonemptiness of the proper efficient solution set and a sufficient condition for the domination property in nonconvex multiobjective optimization problems. A new necessary condition for a point to be efficient or weakly efficient solution is given without any convexity assumption. We also provide a finer outer estimate for the asymptotic cone of the weakly efficient solution set in the quasiconvex case. Finally, we apply our results to the linear fractional multiobjective optimization problem.

Quadratic scalarization for decomposed multiobjective optimization

Practical applications in multidisciplinary engineering design, business management, and military planning require distributed solution approaches for solving nonconvex, multiobjective optimization problems (MOPs). Under this motivation, a quadratic scalarization method (QSM) is developed with the goal to preserve decomposable structures of the MOP while addressing nonconvexity in a manner that avoids a high degree of nonlinearity and the introduction of additional nonsmoothness. Under certain assumptions, necessary and sufficient conditions for QSM-generated solutions to be weakly and properly efficient for an MOP are developed, with any form of efficiency being understood in a local sense. QSM is shown to correspond with the relaxed, reformulated weighted-Chebyshev method as a special case. An example is provided for demonstrating the application of QSM to a nonconvex MOP.

Non-dominated Sorting Disruption-based Gravitational Search Algorithm with Mutation Scheme for Multi-objective Short-Term Hydrothermal Scheduling

The short-term economical/environmental hydrothermal scheduling problem is formulated as a non-linear and non-convex constrained multi-objective optimization problem considering transmission losses and valve point loading effects. This article presents a non-dominated sorting disruption-based gravitational search algorithm with mutation to solve fixed-head and variable-head short-term economical/environmental hydrothermal scheduling problems. In this solution technique, a set of non-dominated solutions are obtained by using the concept of non-dominated sorting and an external archive. Thereafter, a fuzzy decision-making approach has been applied to achieve a suitable and the best compromising solution from the non-dominated solution set. Finally, the non-dominated sorting disruption-based gravitational search algorithm with mutation approach is demonstrated on three test systems: fixed-head two hydro and two thermal plants, two hydro and four thermal plants, and variable-head cascaded four hydro and three thermal plants. Simulation results obtained from this approach are compared with the other methods reported in the literature and it has been found that the proposed approach yields better solutions and is efficient for solving short-term economical/environmental hydrothermal scheduling problems.

A new scalarization method for finding the efficient frontier in non-convex multi-objective problems

One of the most important issues in multi-objective optimization problems (MOPs) is finding Pareto optimal points on the Pareto frontier. This topic is one of the oldest challenges in science and engineering. Many important problems in engineering need to solve a non-convex multi-objective optimization problem (NMOP) in order to achieve the proper results. Gradient based methods, such as Normal Boundary Intersection (NBI), for solving a MOP require solving at least one optimization problem for each solution point. This method can be computationally expensive with an increase in the number of variables and/or constraints of the optimization problem. Nevertheless, the NBI method is a technique motivated by geometrical intuition to provide a better parameterization of the Pareto set than that provided by other techniques. This parameterization is better in the sense that the points obtained by using the NBI method produce a more even coverage of the Pareto curve and this coverage does not miss the interesting middle part of the Pareto curve.This useful property, provides an incentive to create a new method. The first step in this study is using a modified convex hull of individual minimum (mCHIM) in each iteration. The second step is introducing an efficient scalarization problem in order to find the Pareto points on the Pareto front. It can be shown that the corresponding solutions of the MOP have uniform spread and also weak Pareto optimal points. It is notable that the NBI and proposed methods are independent of the relative scale of different objective functions. However, it is quite possible that obtaining a solution of the NBI method not be Pareto optimal (not even locally). Actually, this method aims at getting boundary points rather than Pareto optimal points that will lead to these points which may or may not be a Pareto optimal point. The effectiveness of this method is demonstrated with various test problems in convex and non-convex MOP cases. After that, a few test instances of the CEC 2009 (Zhang et al. 2008) using the proposed method are studied. Also, the relationship between the optimal solutions of the scalarized problem and the Pareto solutions of the multi-objective optimization problem is presented by several theorems.

Branch and probability bound methods in multi-objective optimization

An approach to non-convex multi-objective optimization problems is considered where only the values of objective functions are required by the algorithm. The proposed approach is a generalization of the probabilistic branch-and-bound approach well applicable to complicated problems of single-objective global optimization. In the present paper the concept of probabilistic branch-and-bound based multi-objective optimization algorithms is discussed, and some illustrations are presented.

Reserve Constrained Dynamic Environmental/Economic Dispatch: A New Multiobjective Self-Adaptive Learning Bat Algorithm

This paper proposes a new multiobjective self-adaptive learning bat-inspired algorithm to solve practical reserve constrained dynamic environmental/economic dispatch that considers realistic constraints such as valve-point effects, transmission losses, and ramp rate limits over a short-term time period. Furthermore, to ensure secure real-time power system operations, the system operator must schedule sufficient resources to meet energy demand and operating reserve requirements simultaneously. The proposed problem is a complex nonlinear nonsmooth and nonconvex multiobjective optimization problem whose complexity is increased when considering the above constraints. To this end, this paper utilizes a newly developed meta-heuristic bat inspired algorithm to achieve the set of nondominated (Pareto-optimal) solutions. This algorithm is equipped with a novel self-adaptive learning to increase the population diversity and amend the convergence criteria. The initial population of the proposed framework is generated by a chaos-based strategy. In addition, a tournament crowded selection approach is implemented to choose the population such that the Pareto-optimal front is distributed uniformly, while the extreme points of the tradeoff surface are achieved simultaneously. Numerical results evaluate the performances of the framework for real-size test systems.

A new scalarization and numerical method for constructing the weak Pareto front of multi-objective optimization problems

A numerical technique is presented for constructing an approximation of the weak Pareto front of nonconvex multi-objective optimization problems, based on a new Tchebychev-type scalarization and its equivalent representations. First, existing results on the standard Tchebychev scalarization, the weak Pareto and Pareto minima, as well as the uniqueness of the optimal value in the Pareto front, are recalled and discussed for the case when the set of weak Pareto minima is the same as the set of Pareto minima, namely, when weak Pareto minima are also Pareto minima. Of the two algorithms we present, Algorithm 1 is based on this discussion. Algorithm 2, on the other hand, is based on the new scalarization incorporating rays associated with the weights of the scalarization in the value (or objective) space, as constraints. We prove two relevant results for the new scalarization. The new scalarization and the resulting Algorithm 2 are particularly effective in constructing an approximation of the weak Pareto sections of the front. We illustrate the working and capability of both algorithms by means of smooth and nonsmooth test problems with connected and disconnected Pareto fronts.

Inner and outer estimates for the solution sets and their asymptotic cones in vector optimization

We use asymptotic analysis to develop finer estimates for the efficient, weak efficient and proper efficient solution sets (and for their asymptotic cones) to convex/quasiconvex vector optimization problems. We also provide a new representation for the efficient solution set without any convexity assumption, and the estimates involve the minima of the linear scalarization of the original vector problem. Some new necessary conditions for a point to be efficient or weak efficient solution for general convex vector optimization problems, as well as for the nonconvex quadratic multiobjective optimization problem, are established.

Combined Homotopy Method for a Class of Nonconvex Multiobjective Optimization Problems

Combined Homotopy Method for a Class of Nonconvex Multiobjective Optimization Problems

Nonlinear augmented Lagrangian for nonconvex multiobjective optimization

In this paper, based on the ordering relations induced by a pointed, closed and convex cone with a nonempty interior, we propose a nonlinear augmented Lagrangian dual scheme for a nonconvex multiobjective optimization problem by applying a class of vector-valued nonlinear augmented Lagrangian penalty functions. We establish the weak and strong duality results, necessary and sufficient conditions for uniformly exact penalization and exact penalization in the framework of nonlinear augmented Lagrangian. Our results include several ones in the literature as special cases.

Finding Efficient Solutions by Free Disposal Outer Approximation

The main purpose of this paper is to present a method for generating a representation of efficient solutions and efficient values for nonconvex multiobjective optimization problems. The method is based on a particular outer approximation of nonconvex sets by so-called free disposal nonconvex polyhedra. The convergence of the method is proven, and a discussion on the rate of convergence is addressed with emphasis on the case of two objectives. Examples of up to four objectives are tested, some of which are taken from the recent literature. The numerical experience shows that the optimal value representation obtained by the algorithm is nicely distributed among the set of efficient values of the problem.

Multiobjective bilevel optimization

In this work nonlinear non-convex multiobjective bilevel optimization problems are discussed using an optimistic approach. It is shown that the set of feasible points of the upper level function, the so-called induced set, can be expressed as the set of minimal solutions of a multiobjective optimization problem. This artificial problem is solved by using a scalarization approach by Pascoletti and Serafini combined with an adaptive parameter control based on sensitivity results for this problem. The bilevel optimization problem is then solved by an iterative process using again sensitivity theorems for exploring the induced set and the whole efficient set is approximated. For the case of bicriteria optimization problems on both levels and for a one dimensional upper level variable, an algorithm is presented for the first time and applied to two problems: a theoretical example and a problem arising in applications.

Generating the weakly efficient set of nonconvex multiobjective problems

We present a method for generating the set of weakly efficient solutions of a nonconvex multiobjective optimization problem. The convergence of the method is proven and some numerical examples are encountered.

A Simulated Annealing Algorithm for Prioritized Multiobjective Optimization—Implementation in an Adaptive Model Predictive Control Configuration

This paper presents a new stochastic algorithm for solving hierarchical multiobjective optimization problems. The algorithm is based on the simulated annealing concept and returns a single solution that corresponds to the lexicographic ordering approach. The algorithm optimizes simultaneously the multiple objectives by assigning a different initial temperature to each one, according to its position in the hierarchy. A major advantage of the proposed method is its low computational cost. This is very critical, particularly, for online applications, where the time that is available for decision making is limited. The method is tested in a number of benchmark problems, which illustrate its ability to find near-optimal solutions even in nonconvex multiobjective optimization problems. The results are comparable with those that are produced by state-of-the-art multiobjective evolutionary algorithms, such as the Nondominated Sorting Genetic Algorithm II. The algorithm is further applied to the solution of a large-scale problem that is formulated online, when a multiobjective adaptive model predictive control (MPC) configuration is adopted. This particular control scheme involves an adaptive discrete-time model of the system, which is developed using the radial-basis-function neural-network architecture. A key issue in the success of the adaptation strategy is the introduction of a persistent excitation constraint, which is transformed to a top-priority objective. The overall methodology is applied to the control problem of a pH reactor and proves to be superior to conventional MPC configurations.

Optimality Conditions for the Difference of Convex Set-Valued Mappings

Using the concept of subdifferential of cone-convex set valued mappings recently introduced by Baier and Jahn J. Optimiz. Theory Appl. 100 (1999), 233–240, we give necessary optimality conditions for nonconvex multiobjective optimization problems. An example illustrating the usefulness of our results is also given.

Linear fractional vector optimization problems with many components in the solution sets

Linear fractional vector optimization (LFVO) problems form a special class of nonconvex multiobjective optimization problems which has a significant role both in the management science and in the theory of vector optimization. Up to now, only LFVO problems with at most two connected components in the solution sets have been discussed in the literature. We propose some examples of LFVO problems with three or more connected components in the solution sets. It is proved that for any integer $m$ there exist LFVO problems with $m$ objective criteria whose solution sets have exactly $m$ connected components. Besides, we have solved the conjecture saying that $\chi(E(\mbox{P}))\leq \min\{m,\mbox{dim}0^+D+1\},$ where $\chi(E(\mbox{P}))$ is the number of connected components in the efficient solution set of a LFVO problem $(\mbox{P})$, $m$ is the number of the objective criteria of $(\mbox{P})$, and $\mbox{dim}0^+D$ is the dimension of the recession cone $0^+D$ of the feasible domain $D$ of $(\mbox{P})$. These new facts are useful for analyzing the practical problems which can be modeled as quasiconcave vector maximization problems in general, and as LFVO problems on unbounded feasible domains in particular.

Computing Pareto fronts using distributed agents

Many problems that face a business decision maker are most accurately formulated as multi-objective optimization problems. However, actually solving these problems is a difficult and computationally expensive process. In this paper, we develop and use an agent-based optimization system for efficiently generating the non-dominated solution set to a non-convex multi-objective optimization problem. Through simulation, we highlight the benefits that arise from allowing collaboration among agents embodying a diverse set of algorithms. Further, we demonstrate the ability of the agent system to use large-scale parallel resources efficiently. We also propose two new metrics for quantifying the performance of multi-objective optimization systems.