I have a problem statement where I have to maximize multiple (2 or more than 2) objective functions simultaneously and the some of the variables involved are discrete whereas the remaining are continuous. Is there any julia pacage that can fulfill my needs?
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The JuliaOpt organization is a good place to start for anything optimization-related in Julia. They provide interfaces to a number of standard solvers for linear, mixed and higher-order programs. I’ve never used it, but MultiJuMP.jl looks like something you might be interested in. It doesn’t appear to be part of JuliaOpt, but builds on their interface and model-building interface.
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I have solved some mixed integer (combinatorial) multi-objective optimization problems in the past using Julia and JuMP.
Paper:
http://www.sciencedirect.com/science/article/pii/S0098135416300965
Julia code for all of the examples in the paper:
Where’s the citation to JuMP? 
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Oops. We included the JuMP citation is our subsequent papers.