Constraint Violation Function Research Articles

A novel penalty function-based interval constrained multi-objective optimization algorithm for uncertain problems

To directly address problems involving uncertain objectives and constraints, a novel penalty function-based interval constrained multi-objective optimization algorithm (PF-ICMOA) was developed. It comprises three main components: an uncertainty propagation-based interval constraint solver, an interval constraint violation function, and a two-stage penalty function. The uncertainty violation of interval constraints can be accurately evaluated by the proposed advanced multi-objective optimization algorithm using a segmentation function without requiring multiple interval analyses. Additionally, by applying penalties to infeasible solutions in the two-stage process, partially feasible and superior infeasible solutions are retained in the early stages of evolution, while superior feasible solutions are favored in the later stages. A novel benchmark was designed and the effect of the imprecision factor on the Pareto front was examined in details. Comparisons with seven deterministic constraint handling techniques demonstrated the effectiveness of the proposed method for solving interval constrained multi-objective optimization problems, demonstrating that PF-ICMOA is a highly competitive method for balancing the feasibility, convergence, and diversity performance characteristics.

Constrained stochastic blackbox optimization using a progressive barrier and probabilistic estimates

This work introduces the StoMADS-PB algorithm for constrained stochastic blackbox optimization, which is an extension of the mesh adaptive direct-search (MADS) method originally developed for deterministic blackbox optimization under general constraints. The values of the objective and constraint functions are provided by a noisy blackbox, i.e., they can only be computed with random noise whose distribution is unknown. As in MADS, constraint violations are aggregated into a single constraint violation function. Since all functions values are numerically unavailable, StoMADS-PB uses estimates and introduces so-called probabilistic bounds for the violation. Such estimates and bounds obtained from stochastic observations are required to be accurate and reliable with high but fixed probabilities. The proposed method, which allows intermediate infeasible iterates, accepts new points using sufficient decrease conditions and imposing a threshold on the probabilistic bounds. Using Clarke nonsmooth calculus and martingale theory, Clarke stationarity convergence results for the objective and the violation function are derived with probability one.

An area-type nonmonotone filter method for nonlinear constrained optimization

<abstract><p>In this paper, we define a new area-type filter algorithm based on the trust-region method. A relaxed trust-region quadratic correction subproblem is proposed to compute the trial direction at the current point. Consider the objective function and the constraint violation function at the current point as a point pair. We divide the point pairs into different partitions by the dominant region of the filter and calculate the contributions of the point pairs to the area of the filter separately. Different from the conventional filter, we define the contribution as the filter acceptance criterion for the trial point. The nonmonotone area-average form is also adopted in the filter mechanism. In this paper, monotone and nonmonotone methods are proposed and compared with the numerical values. Furthermore, the algorithm is proved to be convergent under some reasonable assumptions. The numerical experiment shows the effectiveness of the algorithm.</p></abstract>

A Constrained Multiobjective Evolutionary Algorithm With Detect-and-Escape Strategy

Overall constraint violation functions are commonly used in multiobjective evolutionary algorithms (MOEAs) for handling constraints. Constraints could cause these algorithms stuck in two stagnation states: 1) since the feasible region of a multiobjective optimization problem can consist of several disconnected feasible subregions, the search can be easily trapped in a feasible subregion which does not contain all the global Pareto optimal solutions and 2) an overall constraint violation function may have many nonzero minimal points, it can make the search stuck in an unfeasible area. To address these two issues, this article proposes a strategy to detect whether or not the search is stuck in these two stagnation states and then escape from them. Our proposed detect-and-escape strategy uses the feasible ratio and the change rate of overall constraint violation to detect stagnation, and adjusts the weight of the constraint violation for guiding the search to escape from stagnation states. We develop and implement a decomposition-based constrained MOEA with this strategy. Extensive experiments on a number of benchmark problems demonstrate the competitiveness of our proposed algorithm when compared to five other state-of-the-art constrained evolutionary algorithms.

A New QP-free Algorithm Without a Penalty Function or a Filter for Nonlinear Semidefinite Programming

In this paper, we present a QP-free algorithm without a penalty function or a filter for nonlinear semideflnite programming. At each iteration, two systems of linear equations with the same coefficient matrix are solved to determine search direction; the nonmonotone line search ensures that the objective function or constraint violation function is sufficiently reduced. There is no feasibility restoration phase in our algorithm, which is necessary for traditional filter methods. The proposed algorithm is globally convergent under some mild conditions. Preliminary numerical results indicate that the proposed algorithm is comparable.

Translation, solving scheme, and implementation of a periodic and optimal impulsive state control problem

The periodic solution of the impulsive state feedback controls (ISFC) has been investigated extensively in the last decades. However, if the ecosystem is exploited in a period mode, what strategies are implemented to optimize the cost function at the minimal cost? Firstly, under the hypothesis that the system has a periodic solution, an optimal problem of ISFC is transformed into a parameter optimization problem in an unspecified time with inequality constraints, and together with the constraint of the first arrival threshold. Secondly, the rescaled time and a constraint violation function are introduced to translate the above optimal problem to a parameter selection problem in a specified time with the unconstraint. Thirdly, gradients of the objective function on all parameters are given to compute the optimal value of the cost function. Finally, three examples involving the marine ecosystem, computer virus, and resource administration are illustrated to confirm the validity of our approaches.

On a two-phase approximate greatest descent method for nonlinear optimization with equality constraints

Lagrange multipliers are usually used in numerical methods to solve equality constrained optimization problems. However, when the intersection between a search region for a current point and the feasible set defined by the equality constraints is empty, Lagrange multipliers cannot be used without additional conditions. To cope with this condition, a new method based on a two-phase approximate greatest descent approach is presented in this paper. In Phase-Ⅰ, an accessory function is used to drive a point towards the feasible set and the optimal point of an objective function. It has been observed that for some current points, it may be necessary to maximize the objective function while minimizing the constraint violation function in a current search region in order to construct the best numerical iterations. When the current point is close to or inside the feasible set and when optimality conditions are nearly satisfied, the numerical iterations are switched to Phase-Ⅱ. The Lagrange multipliers are defined and used in this phase. The approximate greatest descent method is then applied to minimize a merit function which is constructed from the optimality conditions. Results of numerical experiments are presented to show the effectiveness of the aforementioned two-phase method.

A trust region-based two phase algorithm for constrained black-box and grey-box optimization with infeasible initial point

This paper presents an algorithm for constrained black-box and grey-box optimization. It is based on surrogate models developed using input-output data in a trust-region framework. Unlike many current methods, the proposed approach does not require feasible initial point and can handle hard constraints via a novel optimization-based constrained sampling scheme. A two-phase strategy is employed, where the first phase involves finding feasible point through minimizing a smooth constraint violation function (feasibility phase). The second phase improves the objective in the feasible region using the solution of the feasibility phase as starting point (optimization phase). The method is applied to solve 92 test problems and the performance is compared with established derivative-free solvers. The two-phase algorithm outperforms these solvers in terms of number of problems solved and number of samples used. We also apply the algorithm to solve a chemical process design problem involving highly-coupled, nonlinear algebraic and partial differential equations.

A QP-free algorithm without a penalty function or a filter for nonlinear general-constrained optimization

In this paper, we present a QP-free algorithm without a penalty function or a filter for nonlinear general-constrained optimization. At each iteration, three systems of linear equations with the same coefficient matrix are solved to yield search direction; the nonmonotone line search ensures that the objective function or constraint violation function is sufficiently reduced. There is no feasibility restoration phase in our algorithm, which is necessary for filter methods. The algorithm possesses global convergence as well as superlinear convergence under some mild conditions including a weaker assumption of positive definiteness. Finally, some preliminary numerical results are reported.

Solving engineering optimization problems with tabu/scatter search

This paper introduces an adaptation of a multiobjective tabu/scatter search to deal with nonlinear discrete, mixed-integer constrained engineering optimization problems. The problem is reduced to a bi-objective problem (the objective function and the constraint violation function). This approach eliminates the use of penalties for constraint handling. Its performance was proved with different standard engineering optimization problems, including mathematical function minimization and structural engineering. The results show that the proposed method performs well in terms of efficiency and robustness.

Decentralised coordinated control of microgrid based on multi‐agent system

In this study, a two-hierarchical decentralised coordinated control scheme based on the multi-agent system (MAS) is proposed to improve the security and the stability of the microgrid. The coordinated switching control strategies are constructed in upper level agent to ensure the microgrid security. The making process of the coordinated switching control strategies is regarded as searching the optimal switching operation mode by means of designing constraint violation function and colour Petri-net, so that the operation mode of the microgrid can be switched during larger power unbalance. The decentralised continuous control is designed by integrating the outer loop droop controller with the inner loop prescribed performance controller in the lower level unit agents to guarantee its unit system stability. Finally, the simulation results demonstrate the effectiveness of the decentralised coordination control based on MAS.

AN UNBIASED BI-OBJECTIVE OPTIMIZATION MODEL AND ALGORITHM FOR CONSTRAINED OPTIMIZATION

Transforming a constrained optimization problem (COP) into a bi-objective optimization problem (BOP) is an efficient way to solve the COP. However, how to obtain a good balance between the objective function and the constraint violation function is not easy in BOP. To handle this issue, a novel unbiased bi-objective optimization model is proposed, in which both objective functions are equally treated. Furthermore, the novel model is shown to have the unique Pareto optimal vector under proper condition, and the Pareto optimal vector is exactly corresponding to the optimal solution of COP. Moreover, the relationship between the existing biased bi-objective model and the proposed unbiased one is analyzed in detail. For the unbiased model, a generic multi-objective optimization evolutionary algorithm, i.e. a differential evolution (DE), can be used to solve it, and Pareto ranking is employed as the unique selection criterion. The experiments are conducted on 24 well-known benchmark test instances and the results illustrate that the proposed model is not only effective but also efficient.

Global and local convergence of a nonmonotone SQP method for constrained nonlinear optimization

In this paper, we propose a robust sequential quadratic programming (SQP) method for nonlinear programming without using any explicit penalty function and filter. The method embeds the modified QP subproblem proposed by Burke and Han (Math Program 43:277---303, 1989) for the search direction, which overcomes the common difficulty in the traditional SQP methods, namely the inconsistency of the quadratic programming subproblems. A non-monotonic technique is employed further in a framework in which the trial point is accepted whenever there is a sufficient relaxed reduction of the objective function or the constraint violation function. A forcing sequence possibly tending to zero is introduced to control the constraint violation dynamically, which is able to prevent the constraint violation from over-relaxing and plays a crucial role in global convergence and the local fast convergence as well. We prove that the method converges globally without the Mangasarian---Fromovitz constraint qualification (MFCQ). In particular, we show that any feasible limit point that satisfies the relaxed constant positive linear dependence constraint qualification is also a Karush---Kuhn---Tucker point. Under the strict MFCQ and the second order sufficient condition, furthermore, we establish the superlinear convergence. Preliminary numerical results show the efficiency of our method.

SO-I: a surrogate model algorithm for expensive nonlinear integer programming problems including global optimization applications

This paper presents the surrogate model based algorithm SO-I for solving purely integer optimization problems that have computationally expensive black-box objective functions and that may have computationally expensive constraints. The algorithm was developed for solving global optimization problems, meaning that the relaxed optimization problems have many local optima. However, the method is also shown to perform well on many local optimization problems, and problems with linear objective functions. The performance of SO-I, a genetic algorithm, Nonsmooth Optimization by Mesh Adaptive Direct Search (NOMAD), SO-MI (Muller et al. in Comput Oper Res 40(5):1383---1400, 2013), variable neighborhood search, and a version of SO-I that only uses a local search has been compared on 17 test problems from the literature, and on eight realizations of two application problems. One application problem relates to hydropower generation, and the other one to throughput maximization. The numerical results show that SO-I finds good solutions most efficiently. Moreover, as opposed to SO-MI, SO-I is able to find feasible points by employing a first optimization phase that aims at minimizing a constraint violation function. A feasible user-supplied point is not necessary.

Filter-based genetic algorithm for mixed variable programming

In this paper, Filter Genetic Algorithm (FGA) method is proposed to find the global optimal of the constrained mixed variable programming problem. The considered problem is reformulated to take the form of optimizing two functions, the objective function and the constraint violation function. Then, the filter set methodology [5] is applied within a genetic algorithm framework to solve the reformulated problem. We use pattern search as local search to improve the obtained solutions. Moreover, the gene matrix criteria [10] has been applied to accelerated the search process and to terminate the algorithm. The proposed method FGA is promising compared with some other methods existing in the literature.

Approximate Greatest Descent Methods for Optimization with Equality Constraints

In an optimization problem with equality constraints the optimal value function divides the state space into two parts. At a point where the objective function is less than the optimal value, a good iteration must increase the value of the objective function. Thus, a good iteration must be a balance between increasing or decreasing the objective function and decreasing a constraint violation function. This implies that at a point where the constraint violation function is large, we should construct noninferior solutions relative to points in a local search region. By definition, an accessory function is a linear combination of the objective function and a constraint violation function. We show that a way to construct an acceptable iteration, at a point where the constraint violation function is large, is to minimize an accessory function. We develop a two-phases method. In Phase I some constraints may not be approximately satisfied or the current point is not close to the solution. Iterations are generated by minimizing an accessory function. Once all the constraints are approximately satisfied, the initial values of the Lagrange multipliers are defined. A test with a merit function is used to determine whether or not the current point and the Lagrange multipliers are both close to the optimal solution. If not, Phase I is continued. If otherwise, Phase II is activated and the Newton method is used to compute the optimal solution and fast convergence is achieved.

Optimization of multi-pass turning economies through a hybrid particle swarm optimization technique

Enhancing the performance of manufacturing operations represents a significant goal, especially when cost savings are linked with economies of scale to be exploited. In the area of machining optimization, the selection of optimal cutting parameters subjected to a set of technological constraints plays a key role. This paper presents a novel hybrid particle swarm optimization (PSO) algorithm for minimizing the production cost associated with multi-pass turning problems. The proposed optimization technique consists of a PSO-based framework wherein a properly embedded simulated annealing (SA), namely an SA-based local search, aims both to enhance the PSO search mechanism and to move the PSO away from being closed within local optima. In order to handle the numerous constraints which characterize the adopted machining mathematical model, a constraint violation function integrated with a suitable objective function has been engaged. In addition, a twofold strategy has been implemented to manage the equality constraint between the provided total depth of cut and the number of passes to be performed. Firstly, an accurate problem encoding involving only five cutting parameters has been performed. Secondly, a proper repair procedure that should be run just before any solution evaluation has been engaged. Five different test cases based on the multi-pass turning of a bar stock have been used for comparing the performance of the proposed technique with other existing methods.

Using "Filter" Approach to Solve the Constrained Optimization Problems

In this paper, the solution of constrained nonlinear programming problems by a Sequential Quadratic Programming (SQP) is considered. The aim of the present work is to promote global convergence without the need to use a penalty and Barrier functions in the mixed interior-exterior point method. Instead, a new concept of a “filter” that aims to minimize the objective function and its approach that allows appoint to be accepted if reduces the objective function and satisfies the constraint violation function. If that point is rejected a new point is tested.

A globally convergent trust region multidimensional filter SQP algorithm for nonlinear programming

We propose a trust region multidimensional filter SQP algorithm. The multidimensional filter technique proposed by Gould et al. [SIAM J. Optim., 15 (2005), pp. 17–38] is extended to solve constrained optimization problems. The constraints are partitioned into p parts. The entry of our filter consists of these different parts. Not only the criteria for accepting a trial step would be relaxed, but also the individual behaviour of each part of the constraints is considered. The filter's entries and the acceptance criteria are different from other filter-related algorithms in the literature. It should be noted that the undesirable link between the objective function and the constraint violation function in the filter acceptance criteria disappears. Our algorithm is also combined with the non-monotone technique for accepting a trial step, which leads to a more flexible acceptance criteria. Under mild conditions, global convergence is proved. Numerical results show the robustness and efficiency of our algorithm.

A Progressive Barrier for Derivative-Free Nonlinear Programming

We propose a new constraint-handling approach for general constraints that is applicable to a widely used class of constrained derivative-free optimization methods. As in many methods that allow infeasible iterates, constraint violations are aggregated into a single constraint violation function. As in filter methods, a threshold, or barrier, is imposed on the constraint violation function, and any trial point whose constraint violation function value exceeds this threshold is discarded from consideration. In the new algorithm, unlike the filter method, the amount of constraint violation subject to the barrier is progressively decreased adaptively as the iteration evolves. We test this progressive barrier (PB) approach versus the extreme barrier (EB) with the generalized pattern search (Gps) and the lower triangular mesh adaptive direct search (LTMads) methods for nonlinear derivative-free optimization. Tests are also conducted using the Gps-filter, which uses a version of the Fletcher–Leyffer filter approach. We know that Gps cannot be shown to yield kkt points with this strategy or the filter, but we use the Clarke nonsmooth calculus to prove Clarke stationarity of the sequences of feasible and infeasible trial points for LTMads-PB. Numerical experiments are conducted on three academic test problems with up to 50 variables and on a chemical engineering problem. The new LTMads-PB method generally outperforms our LTMads-EB in the case where no feasible initial points are known, and it does as well when feasible points are known. which leads us to recommend LTMads-PB. Thus the LTMads-PB is a useful practical extension of our earlier LTMads-EB algorithm, particularly in the common case for real problems where no feasible point is known. The same conclusions hold for Gps-PB versus Gps-EB.