Optimization of random cost functions and statistical physics

This is the text of my report presented at the 29th Solvay Conference on Physics on `The Structure and Dynamics of Disordered Systems' held in Bruxelles from October 19 to 21, 2023. I consider the problem of minimizing a random energy function $H(\sigma)$, where $\sigma$ is an $N$-dimensional vector, in the high-dimensional regime $N\gg 1$. Using as a reference point a 1986 paper by Fu and Anderson, I take stock of the progress on this question over the last 40 years. In particular, I focus on the influence and ramifications of ideas originating from statistical physics. My own conclusion is that several of the most fundamental questions in this area (which in 1986 were barely formulated) have now received mathematically rigorous answers, at least in simple -- yet highly nontrivial -- settings. Instrumental to this spectacular progress was the dialogue between different research communities: physics, computer science, mathematics.