Convex multi-objective optimization

Multi-objective optimization (MOO), also known as multi-criteria optimization, particularly outside engineering, refers to finding values of decision variables which correspond to and provide the optimum of more than one objective. Unlike in SOO which gives a unique solution (or several multiple optima such as local and global optima in case of non-convex problems), there will be many optimal solutions for a multiobjective problem the exception is when the objectives are not conflicting in which case only one unique solution is expected. Hence, MOO involves special methods for considering more than one objective and analyzing the results obtained. 

Metev, B. S., and Yordanova-Markova, 1. T. (1997), Multi-objective Optimization over Convex Disjunctive Feasible Sets Using Reference Points, European Journal of Operational Research, Vol. 98, pp. 124-137. 

Another consideration concerns the shape of the fronts. They appear convex, and would not be determinable with conventional multi-objective optimization methods like, for example, the Weighted Sum Method. This last method is a linear combination of the objectives, and the application of an Evolutionary optimization problem approach in Pareto s fronts definition would be more appropriate. 

In this chapter, the integrated design and control of bioprocesses was considered as a multi-objective optimization problem subject to non-linear differential-algebraic constraints. This formulation has a number of advantages over the traditional sequential approach, not only because it takes into account the process dynamics associated to a particular design, but also it provides a set of possible solutions from which the engineer can choose the most appropriate to his/her requirements. However, these problems are usually challenging to solve due to their non-convexity, which causes the failure of procedures based on local (e.g. SQP) NLP solvers. 

A short list of such tools is reported in Table 1, where GUI and MOO indicate, respectively, if the software has a Graphical User Interface and if it allows for multi-objective optimization. We also indicate if each tool is open-source and if it allows for customization of the optimization algorithms. We should note that we included in the table only multi-disciplinary optimization tools that are specifically based on CIO methods, while we excluded software based on classic techniques for convex optimization, integer linear/non-linear programming and methods addressing combinatorial optimization only. We also excluded those technical software products that are not devoted specifically to optimization but still may include optimization methods, such as Matlab (which provides the Optimization Toolbox), and other CAD/CAE or multi-physics software, as well as multi-disciplinary tools that provide (as an extra feature) one or more, often domain-specific, optimization techniques (see for instance AVL CAMEO [23]). 

In this paper, we extend the work of [10] by simultaneously considering minimization of the total utility consumption, maximization of operational flexibility to source-stream temperatures, and even minimum number of matches as multiple design objectives. The flexible HEN synthesis problem is thus formulated as the one of multi-objective mixed-integer linear programming (MO-MILP). This formulation also assumes that the feasible region in the space of uncertain input parameters is convex, so that the optimal solution can thus be explored on the basis of the vertices... 

To solve the above optimization problem, a Multiple-Objective Evolutionary Algorithms (MOEA) is embraced here. MOEA is a term employed in the Evolutionary Multi-criteria Optimization field to refer to a family of evolutionary algorithms formulated to deal with MO. MOEA are able to deal with non-continuos, non-convex and/or non-linear objectives/constraints, and objective functions possibly not explicitly known (e.g. the output of Monte Carlo simulation tuns). 

Unconstrained optimization deals with situations where the constraints can be eliminated from the problem by substitution directly into the objective function. Many optimization techniques rely on the solution of unconstrained subproblems. The concepts of convexity and concavity will be introduced in this subsection, as well as discussing unimodal versus multimodal functions, singlevariable optimization techniques, and examining multi-variable techniques. 

Analytical studies for choosing the parameters of due date rules include [88] and [108]. Seidmann and Smith [88] consider the CON due date policy in a multi-machine environment under the objective of minimizing the weighted sum of earliness, tardiness and lead time penalty (the same objective function is also considered in [87]). They assume that the shop is using a priority discipline for sequencing, such as FCFS or EDD, and the probability distribution of the flow time is known and is common to all jobs. They show that the optimal lead time is a unique minimum point of strictly convex functions can be found by simple numerical search. 

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