Motivated by recent work of Renegar, we present new computational methods and associated computational guarantees for solving convex optimization problems using first-order methods. Our problem of interest is the general convex optimization problem f^* = \min_{x \in Q} f(x), where we presume knowledge of a strict lower bound f_slb < f^*. [Indeed, f_slb is naturally known when optimizing many loss functions in statistics and machine learning (least-squares, logistic loss, exponential loss, total variation loss, etc.) as well as in Renegar's transformed version of the standard conic optimization problem; in all these cases one has f_slb = 0 < f^*.] We introduce a new functional measure called the growth constant G for f(.), that measures how quickly the level sets of f(.) grow relative to the function value, and that plays a fundamental role in the complexity analysis. When f(.) is non-smooth, we present new computational guarantees for the Subgradient Descent Method and for smoothing methods, that can improve existing computational guarantees in several ways, most notably when the initial iterate x^0 is far from the optimal solution set. When f(.) is smooth, we present a scheme for periodically restarting the Accelerated Gradient Method that can also improve existing computational guarantees when x^0 is far from the optimal solution set, and in the presence of added structure we present a scheme using parametrically increased smoothing that further improves the associated computational guarantees.

## Citation

MIT Operations Research Center Working Paper, MIT, September 2015.