Optimization

I recently read the paper Carbon-Aware Computing for Data Centers with Probabilistic Performance Guarantees which demonstrated the benefits and power of a Distributionally Robust Optimization method and I wanted to dive deeper into the method. This blog post is a result of the deep dive. Introduction In Stochastic Optimization (SO), we are interested in solutions that account for different possible outcomes of uncertain data. Accounting for this uncertainty requires us to define a probability distribution over the uncertain quantities—a forecast error, a demand signal, a compute load—and then optimize an objective defined with respect to that distribution. The choice of how to use the distribution depends on the application: one might minimize expected cost, guard against a high-percentile loss using Conditional Value-at-Risk (CVaR), or enforce a chance constraint that a capacity limit is violated with probability no greater than some threshold. What all of these approaches share is the assumption that we know, or can accurately estimate, the distribution itself. Distributionally Robust Optimization is the framework we reach for when that assumption breaks down. ...

Many applications of optimization and control are performed in a deterministic setting. That is, the quantities of the problem such as the state variables, control variables, and or parameters of a model are treated as fixed known values. This assumption may be reasonable in applications such as robotics, but in others such as resource allocation, we may need to incorporate the uncertainty of various quantities in order to compute the expectation of the objective. ...

Uber allocates money across different regions and programs to incentivize riders and drivers to use Uber products. This incentive structure ultimately influences the market. This leads to the natural question of “how much to allocate to each city and which program” to maximize business objectives? Uber has a finite amount of money that must be allocated accordingly. Given a total budget of say, $1,000,000, how should it be divied up amongst the cities and programs? ...

#| code-fold: true import matplotlib.pyplot as plt import torch import numpy as np from botorch.acquisition import qLogExpectedImprovement from botorch.fit import fit_gpytorch_model from botorch.models import SingleTaskGP from botorch.optim import optimize_acqf from gpytorch.mlls import ExactMarginalLogLikelihood from torch.distributions import Normal plt.style.use("https://raw.githubusercontent.com/GStechschulte/filterjax/main/docs/styles.mplstyle") Outcome constraints In optimization, it is often the goal that we need to optimize an objective function while satisfying some constraints. For example, we may want to minimize the scrap rate by finding the optimal process parameters of an manufacturing machine. However, we know the scrap rate cannot be below 0. In another setting, we may want to maximize the throughput of a machine, but we know that the throughput cannot exceed the maximum belt speed of the machine. Thus, we need to find regions in the search space that both yield high objective values and satisfy these constraints. In this blog, we will focus on inequality outcome constraints. That is, the domain of the objective function is ...