Constrained optimal control problem

Question

I am trying to solve an optimization problem which is probably reminiscent of optimal control theory but all of this is not exactly my field of expertize and I am a little bit lost in translation. If someone could please put me on the right tracks, it would most certainly save me a lifetime of useless wanderings.

In discrete time $t=[1,2,...,T]$, I would like to find the vector $\{b_t\}_{t \in T}$ such that $F(b_t):=\sum_{t=1}^{T} h(d_t-b_t)b_t$ is maximized, where:

1. $\{d_t\}_{t \in T}$ is given and $d_t \geq0 \quad \forall t\in T$.
2. $h(\cdot):\mathbb R^+ \to \mathbb R^+ $ is a strictly convex function.
3. $d_t-b_t\geq0 \quad \forall t\in T$
4. $\mid b_t \mid \leq \zeta \quad \zeta \in \mathbb R^+$
5. $\sum_{t=1}^{t'} \big(\theta(b_t)+\eta\theta(-b_t)\big)\cdot b_t\leq 0 \quad \forall t'=[1,2,...,T]$ , where $0<\eta \leq1$ is fixed and $\theta(\cdot)$ is the Heaviside function.

I am pretty novice in optimization theory but from what I have understood this problem could maybe be solved using Karush-Khun-Tucker conditions, but then I don't clearly see how to proceed, notably with the heaviside's. Is there a better approach or is this problem simply unsolvable? Is there good (introductory) literature I should read about this?

Thank you for your help!

Answer 1

I assume that $\theta(y) = 1$ if $y\geq 0$, and $\theta(y)=0$ else. In that case you can "linearize" your constraint 5 as follows: Add new variables $y_t$ for $t \in T$. Replace constraint 5 with the following three linear constraints:

i) $y_t \geq b_t$ for all $t \in T$.

ii) $y_t \geq \eta b_t$ for all $t \in T$.

iii) $\sum_{t=1}^{t'} y_t \leq 0$ for all $t' \in \{1, ..., T\}$.

This works precisely because $0 < \eta \leq 1$. The entire problem would be a convex optimization problem if the functions $h(d_t-b_t)b_t$ were concave in $b_t$ for all $t$.

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Without concavity of the objective function, general solutions seem hard. However, the above transformation at least maps it to a problem of maximizing a (nonconcave) objective function over a convex set defined by linear inequality constraints. And KKT conditions may be easier to state now.