# From constrained to unconstrained maximization problem

I have the following constrained maximization problem $$\max_{X_1,X_2,...,X_i,...,X_N} \sum_{i=1}^{N}X_i f_i(X_1,...,X_N) \hspace{0.2 cm} \text{subject to} \sum_{i=1}^{N}X_i-B\leq 0 \text{ and } X_i\geq 0 \text{ } \forall i=1,...,N$$ where $f_i(X_1,...,X_N)$ is a function of $X_1,...,X_N$ different across $i$ such that the unconstrained maximum of the objective function always exists and $B>0$ .

Do you know whether there is a way to transform it in an unconstrained maximization problem?

• Do you have any reason to believe this is convex? – Michael Grant Dec 25 '14 at 23:23
• It is always possible to transform a constrained optimization problem to an unconstrained one---i.e., using indicator functions. But I suspect that you have a practical purpose in mind that indicator functions are unlikely to satisfy. Perhaps you would like to clarify your true objective here. Are you just looking for a way to solve the problem, and you're assuming that it will be easier if it is unconstrained? Or do you have another important reason why it must be unconstrained? – Michael Grant Dec 26 '14 at 2:28
• @MichaelGrant When you say "It is always possible to transform a constrained optimization problem to an unconstrained one", are you referring to esterior penalty function methods or barrier function methods or multiplier methods? – STF Jan 4 '15 at 16:08
• None of the above. I am referring to the use of indicator functions: functions which are zero on a convex set at $+\infty$ outside of it. These can be added to the objective in place of corresponding constraints, (or in the case of maximization, subtracted), and preserve exact mathematical equivalence, unlike barrier or penalty functions. – Michael Grant Jan 4 '15 at 16:11