Finding trade-off solutions close to KKT points using evolutionary multi-objective optimization

Abstract

Abstract-Despite having a wide-spread applicability of evolutionary optimization procedures over the past few decades, EA researchers still face criticism about the theoretical optimality of obtained solutions. In this paper, we address this issue for problems for which gradients of objectives and constraints can be computed either exactly, or numerically or through subdifferentials. We suggest a systematic procedure of analyzing a representative set of Pareto-optimal solutions for their closeness to satisfying Karush-Kuhn-Tucker (KKT) points, which every optimal solution must also satisfy. The procedure involves either a least-square solution or an optimum solution to a set of linear system of equations involving Lagrange multipliers. The procedure is applied to a number of differentiable and non-differentiable test problems and to a highly nonlinear engineering design problem. The results clearly show that EAs are capable of finding solutions close to theoretically optimal solutions in various problems. As a by-product, the error metric suggested in this paper can also be used as a termination condition for an EA application. Hopefully, this study will bring EAs and its research closer to classical optimization studies.

Reactive power planning with voltage stability enhancement using covariance matrix adapted evolution strategy

“…Three inequality constraints of the load bus voltages (state variable) are active. The number of equations is more than unknowns, then the error is measured by Equation (6) of Reference [17]. The KKT-error for the Case-1 is 0.0377, which is close to zero.…”

“…The obtained results are compared with real-coded GA (RGA), PSO, Broyden-FletcherGoldfarb-Shanno (BFGS), and sequential quadratic programming (SQP) techniques. Recently, Deb et al [17] have proposed a systematic procedure of analyzing the obtained optimal solutions for their closeness to Karush-Kuhn-Tucker (KKT) points, which every optimal solution must satisfy. To confirm a claim on optimality, KKT conditions are also applied to the obtained optimal solutions of CMAES algorithm.…”

Modelling the Pareto-optimal set using B-spline basis functions for continuous multi-objective optimization problems

“…If the objective function is not smooth and there are sudden kinks in the objective function or constraints then the process will have difficulties in reducing error, leading to a higher unacceptable fitting error. In such cases, the sub-differential based KKT-error minimization procedure due to Deb et al (2007) can be adopted.…”

Low power FIR filter design using modified multi-objective artificial bee colony algorithm

“…In this subsection the convergence of the proposed modified multi-objective ABC has been proved analytically on the basis of Karush-Kuhn-Tucker (KKT) optimality conditions (Deb et al, 2007). …”