**1. Introduction**

Many engineering design problems require simultaneous optimization of multiple, often competing, objectives. Unlike in single-objective optimization, a multiobjective problem with competing objectives has no single solution. An optimum solution with respect to only one objective may not be acceptable when measured with respect to the other objectives. Multi-objective problems have a number of solutions called the Pareto-optimal set, named after Vilfred Pareto [1], that represent the range of best possible compromises amongst the objectives. Traditional gradient-based optimization algorithms are capable of addressing the multiobjective problems by converting the problem into a single-objective formulation.

On the other hand, evolutionary algorithms (EAs)<sup>1</sup> are well suited for the multiobjective problems as they can evolve to a set of designs that represent the Pareto frontier in a single run of the algorithm [2, 3]. As a result, EAs often find application to address multi-objective problems. Despite the popularity of these algorithms to solve a wide range of problems, they, like all non-gradient meta-heuristic searches, have issues with computational cost and rate of convergence to the Pareto frontier. After some number of generations, the candidate solutions may begin to exhibit little or no improvement. Modified versions of these algorithms exist which improve the convergence rate [4]. However, hybridizing EAs with an efficient gradient-based algorithm may significantly improve the convergence rate and has demonstrated the ability to solve multi-objective problems more efficiently than the EA alone [3]. Hybridization of an EA with a gradient-based local search algorithm has started to gain popularity owing to its promising capabilities to address the demerits of many optimization algorithms when used independently.

The genetic algorithm (GA) [5] is a class of EA and is a well-known populationbased global search algorithm. Apart from its ability to explore the design space, GA is also capable of handling both discrete and continuous type design variables. This makes the GA an ideal choice to address problems that combine both discrete and continuous variables. However, the GA, like other EAs, does not provide any proof of convergence, and the GA cannot directly enforce constraints. Commonly, constraint handling for a GA search uses a penalty approach such that the fitness function reflects the objective function value and accounts for violated constraints. This generally requires the use of penalty multipliers to adjust the "strength" with which the penalty impacts the fitness function and selecting suitable penalty multipliers is often difficult. Further, for multi-objective problems, the different scaling or magnitude of the objectives can complicate selecting appropriate penalty multipliers.

On the other hand, Sequential Quadratic Programming (SQP) [6], is a wellknown gradient-based search algorithm that directly handles constraints and provides proof of convergence to local optima using Karush-Kuhn-Tucker (KKT) optimality criteria [7]. Because SQP uses gradient information, it is a computationally efficient search algorithm. However, SQP cannot handle discrete design variables or discontinuous functions and has difficulty with multi-modal functions. Therefore, both of these (GA and SQP) well-known optimization algorithms have their own pros and cons that limit their individual applicability to fully address constrained multi-objective problems that combine both continuous and discrete type design variables. Combining the GA with SQP creates a hybrid approach that improves the overall optimization process for constrained mixed-discrete nonlinear programming problems (MDNLP).

The chapter presents a combination of the two-branch tournament GA for multi-objective problems with an SQP-based local search implementation of the goal attainment problem formulation allowing an improved information sharing between the two algorithms. To the best of the authors' knowledge, there exists no work that emphasizes the process of hybridization combining an N-branch tournament selection GA with the goal attainment formulation as the local search in a compatible manner and then demonstrates application of the approach to solve a hard-to-solve constrained multi-objective, mixed-discrete nonlinear optimization problem. Later in the chapter, the hybrid approach is applied to solve a three-bar truss problem, a ten-bar truss problem, and a greener aircraft design optimization

<sup>1</sup> Here, the term "evolutionary algorithm" encompasses all population-based search algorithms that use features inspired by biological evolution.

problem – all representatives of constrained multi-objective, mixed-discrete nonlinear programming problem. The truss problems have basis in test problems for structural optimization, and the motivation to select a greener aircraft design optimization problem arises from the increased concern about the environmental impact of the growing air transportation system.
