How can you develop an algorithm for mixed integer optimization with time-dependent constraints?

The model needs the time dimension. This means that the decision variables ad well as the constraints are time dependent. The problem formulation should be general enough to dictate how input data and decision variables interact at time t. The time horizon will determine for how many or which time interval the model should be valid. A classic example in power system studies is : Generation (t) = Demand (t) the t can be expressed in hour,minue or a fraction of seconds.

Time constraints create situations for dynamic solutions because time is the most valuable commodity to man. Focus must be put towards the actual answer to the problem that is defined. Utilizing foundational knowledge on top of the data given at hand is pivotal to creating a solution that satisfies the requirements at hand. From that point applying models that are sophisticated enough to satisfy the problem at hand will provide a solution. After this, seek refinement in all areas where the capability exposes itself.

At times, hybrid approaches combining GAs with other methods might be more suitable. So, I suggest to explore this option as well.

A possible framework is to formulate your problem as an Optimal Control Problem with Time-Dependent Mixed-Integer Constraints. It combines continuous and discrete decisions (control variables) over time. In mathematical terms, and in the context of a deterministic system, something similar to this: Minimize ∫L(x(t), u(t), v(t), t) dt Subject to: x?(t) = f(x(t), u(t), v(t), t), x(t_0) = x_0 u(t) ∈ U ? ?^m v(t) ∈ V ? ?^p S(x(t), u(t), v(t), t) ≤ 0 with: x(t): State variables (continuous) u(t): Continuous control variables v(t): Discrete control variables (integer-valued) f: State dynamics equation L: Lagrangian running cost function U: Continuous control constraints V: Integer control constraints S: Path constraints

If it is possible to solve a problem within the framework of time discretization (up to minutes, hours, days) and the task will not be huge, then we can consider time as one of the sets of task indices. Constraint generation will be standard within mixed integer programming. If it is not possible to use discrete time, but it must be continuous, then such problems are solved using simulation modeling and heuristic algorithms. The restrictions themselves will be modeled by the penalty function in the heuristic.

Choosing the right model for mixed integer optimization with time-dependent constraints involves considering problem complexity, desired solution accuracy, computational resources, and solver availability. Linear problems use MILP models, while non-linear ones require MINLP. Exact methods like Branch and Bound are precise but slow, making heuristic methods like Genetic Algorithms or Tabu Search appealing for quicker, near-optimal solutions. Hybrid approaches can offer a balance between speed and accuracy. Time-dependent constraints necessitate careful formulation, often with time-indexed variables. The decision between models hinges on balancing the nature of the problem, computational limitations, and solution needs.

Instead of implementing your own approach, consider using a "solver" (like InsideOpt Seeker) which is optimization terminology for a program that takes in a declarative optimization model that will be optimized automatically. "Declarative" means that you only need to specify which properties you want your solution to have, usually by specifying three parts: 1. What do you need to decide? In an optimization model, you declare "variables" for this purpose. While the term is the same, a variable in optimization is something very different than in ML or stats: It is the very thing we let the solver *set* for us. 2. What plans are legal? You define side constraints that express what can be executed. 3. What metric do you need to optimize?

If it is possible to solve a problem within the framework of time discretization (up to minutes, hours, days) and the task will not be huge, then we can consider time as one of the sets of task indices. Constraint generation will be standard within mixed integer programming. If it is not possible to use discrete time, but it must be continuous, then such problems are solved using simulation modeling and heuristic algorithms. The restrictions themselves will be modeled by the penalty function in the heuristic.

It looks like mixed integer program with binary variables deciding the sequence and continuous variables deciding the visit times. TSP being an NP hard problem, one might have to switch to heuristic approaches if the problem size is large.

I suggest to explore heuristic models as well. As per my experience, genetic algorithm based model suits better for some optimisation problems at hand. Take final call based on the problem statement and the model compatibility (performance, complexity, compute resource..)