I want to optimize following objective functionmax f = profit(x,y) - expense(x,y) subject to: 0<= x, y <=1 using strength pareto evolutionary algorithm (SPEA2). The objective function is non-linear and is not convex or concave function of decision variables. Can I break objective function into two that is maximize profit(x,y) and minimize expense(x,y) and then optimize them in combination at the end. I do not know if it make some sense, sorry I am completely new to the filed. I shall appreciate any help.
Note: (in general) profit = income - expense; and what you've asked looks very dodgy (e.g. something = profit - expense = income - 2 * expense), so I'm going to assume you meant "income" everywhere you've said "profit".
No, you can't find max. income(x,y) and min. expense(x,y) and combine/optimise them at the end because your can expect a relationship between income and expense (e.g. as expense increases income increases).
Also; don't forget that the best way to approach this kind of problem is to expand and simplify the resulting function.
For a very simple example:
income = items * bonus + items * 0.9 * $123.45
expense = bonus * $1 + items * $99.00
profit = income - expense
= (items * bonus + items * 0.9 * $123.45) - (bonus * $1 + items * $99.00)
= items * bonus - bonus * $1 + items * 0.9 * $123.45 - items * $99.00
= (items - 1) * bonus + items * (0.9 * 123.45 - 99.00)
= (items - 1) * bonus + items * 12.105
In other words; if you could find max. income(x,y) and min. expense(x,y) and combine/optimise them at the end, you still wouldn't want to because it's less efficient/slower, and better/faster to just find max. profit(x,y).