Convex Optimization: Is convexity of the constraint necessary for a correct analytical solution?

Question

I am from the field of economics. For an agent's utility maximization it's common to use convex optimization following the lagrange maultiplier method. This is mostly used to get an analytical result; "The effect of variable $a$ on the value of the variable $X$ that satisfies optimality"

Regarding non-convex constraints, for example for maximizing utility $U$

$$U = α_c \log (c) + α_n \log (n) + α_q \log (q)$$

constrained by: $$c + nb_0 + sn ≤ wl_w$$

where

$$s = f(q)$$

Hence

$$c + nb_0 + f(q)n ≤ wl_w$$

according to the authors (Jones et al, 2008, p.29) the non-linear term $f(q)n$ makes the constraint non-convex. They proceed to apply the "Change of variable" technique to convexify the constraint.

My question: In several set-ups the change of variable technique is an algebraic hassle, which made me think; if we are not seeking a numericl solution anyway, wouldn't solving the initial problem suffice to explore the dynamics that lead to the solution? I am not a mathematician by any stretch of the imagination, so I am anxious that I am missing something obvious.

Citation: Jones, Larry E., Alice Schoonbroodt, and Michele Tertilt. Fertility theories: can they explain the negative fertility-income relationship?. No. w14266. National Bureau of Economic Research, 2008.

Answer 1

This particular utility function $U(c, n, q)$ is monotonically increasing in each of its arguments, so it would be maximized as $c \rightarrow \infty$, $n \rightarrow \infty$, and $q \rightarrow \infty$. In other words, this utility function is not particularly interesting without the constraint. And the unconstrained problem doesn't tell you much about the optimal $c, n, q$ for the constrained problem.

This is in general true of most utility maximization problems. The constraint imposes a tradeoff between different things that humans would like $\infty$ of, and understanding behavior in light of the tradeoffs is exactly what economists seek to understand.

To state this more mathematically, that the constraint matters to the solution of the problem is equivalent to stating that the solution will lie on the boundary of the constraint region, or equivalently, that at the solution, the inequality will be tight (ie that the left-hand side and right-hand side will be equal).

This in turn means that how one "convexifies" the constraint really matters to the solution. If one performs a simple change of variable, then the solution will not change. If, on the other hand, one actually changes the constraint via "convex relaxation" (ie replacing the constraint with a similar convex constraint), it is possible that the solution will be different.

Answer 2

I re-read your question multiple times, and am still unconfortable with the kind of answer you expect. Let's try anyway.

First, we have two exclusive cases:

• Either the change of variable modifies at least slightly the optimal point, and then the conclusion would be mathematically dubious.
• Or the change of variable does not modify the optimal point, and then solving the initial problem will of course give the same result.

Quickly perusing the 78-pages paper you referenced, this is the second case here. So the obvious answer to your question, is: yes, solving the initial problem would suffice to explore the dynamics.

So, why do the authors indulge in ensuring convexity, that does not seem to buy them anything, you may say?

Well, probably because (strict) convexity is what ensures that the function "optimal value depending from such and such variables" is well defined.

Without an argument such as convexity, we could have multiple solutions, or even no solution in case the domain was not compact, or with pathological functions (= discontinuous).

When reasoning about non-existent mathematical objects, it is frequent that anything and its contrary can be deduced. One historical case is the series $1-1+1-1+1-1+...$. There is a kind of "ex falso, quodlibet" effect.

Still, some people would say "OK, but we are modeling a physical reality here, so of course there is a unique solution". Well, if our mathematical model is correct, yes. But if it is false?

Therefore, checking that the mathematical model has one and only one solution for any tuple of entry variables is vital. Otherwise a reader could easily destroy the paper by remarking that in a corner case, the model does not give a unique solution.

One famous (still open) such problem is linked to Navier-Stokes equations for fluids: whether those partial differential equations always solve into smooth solutions for every initial conditions. Non-smooth solutions would be deemed "not physical", so it would be a good testimony that something beyond our current understanding is happening. This is one of the 7 "Millenium problems" from Cray Institute, and the subject of the 2017 film "Mary".

I don't expect to have answered the whole of your question, so please comment as necessary