In this paper, we present a convex optimization framework to compute guaranteed upper bounds on the Lipschitz constant of DNNs both accurately and efficiently.

(Bertsekas, 1999). Algorithm 1. Furthermore, we call ˆ f (), X We can show that | f () ˆ f () |, 8 2 [, ] . Besides, computing the upper bound claimed in Proposition 4.2 requires finding The second equality is from the fact that the objective function is affine w.r.t. Finally, we verify the rest two components. Finally, we verify the rest two components. This finishes the proof of our claim.

However, they bring their own set of challenges.

With our proposed method, decision-makers can simultaneously have the evaluated solutions and the approximate Pareto set.

Biotechnology can benefit from dynamic control to improve production efficiency. In this context, optogenetics enables modulation of gene expression using light as an external input, allowing fine-tuning of protein levels to unlock dynamic metabolic control and regulation of cell growth. Optogenetic systems can be actuated by light intensity. However, relying solely on intensity-driven control (i.e., signal amplitude) may fail to properly tune optogenetic bioprocesses when the dose-response relationship (i.e., light intensity versus gene-expression strength) is steep. In these cases, tunability is effectively constrained to either fully active or fully repressed gene expression, with little intermediate regulation. Pulse-width modulation, a concept widely used in electronics, can alleviate this issue by alternating between fully ON and OFF light intensity within forcing periods, thereby smoothing the average response and enhancing process controllability. Naturally, optimizing pulse-width-modulated optogenetics entails a switching-time optimal control problem with a binary input over many forcing periods. While this can be formulated as a mixed-integer program on a refined time grid, the number of decision variables can grow rapidly with increasing time-grid resolution and number of forcing periods, compromising tractability. Here, we propose an alternative solution based on reinforcement learning. We parametrize control actions via the duty cycle, a continuous variable that encodes the ON-to-OFF switching time within each forcing period, thereby respecting the intrinsic binary nature of the light intensity.

We consider the optimization of an uncertain objective over continuous and multidimensional decision spaces in problems in which we are only provided with observational data.