Shortfall Risk Models When Information on Loss Function Is Incomplete

Utility-based shortfall risk measures effectively captures a decision maker's risk attitude on tail losses. In this paper, we consider a situation where the decision maker's risk attitude toward tail losses is ambiguous and introduce a robust version of shortfall risk, which mitigates the risk arising from such ambiguity. Specifically, we use some available partial information or subjective judgement to construct a set of plausible utility-based shortfall risk measures and define a so-called preference robust shortfall risk as through the worst risk that can be measured in this (ambiguity) set. We then apply the robust shortfall risk paradigm to optimal decision-making problems and demonstrate how the latter can be reformulated as tractable convex programs when the underlying exogenous uncertainty is discretely distributed.

Robustly decorrelating errors with mixed quantum gates

Coherent errors in quantum operations are ubiquitous. Whether arising from spurious environmental couplings or errors in control fields, such errors can accumulate rapidly and degrade the performance of a quantum circuit significantly more than an average gate fidelity may indicate. As shown by Hastings [1] and Campbell [2], by replacing the deterministic implementation of a quantum gate with a randomized ensemble of implementations, one can dramatically suppress coherent errors. Our work begins by reformulating the results of Hastings and Campbell as a quantum optimal control problem. We then discuss a family of convex programs able to solve this problem, as well as a set of secondary objectives designed to improve the performance, implementability, and robustness of the resulting mixed quantum gates. Finally, we implement these mixed quantum gates on a superconducting qubit and discuss randomized benchmarking results consistent with a marked reduction in the coherent error. [1] M. B. Hastings, Quantum Information & Computation 17, 488 (2017). [2] E. Campbell, Physical Review A 95, 042306 (2017).

Maximum Penalty Function in Linear Programming

A linear program can be equivalently reformulated as an unconstrained nonsmooth minimization problem, whose objective is the sum of the original objective and a penalty function with a sufficiently large penalty parameter. The article presents two methods for choosing this parameter. The first one applies to linear programs with usual linear inequality constraints. Then, we use a corresponding theorem by N.Z. Shor on the equivalence of a convex program to an unconstrained nonsmooth minimization problem. The second method is for linear programs of a special type. This means that all inequalities are of the form that a linear expression on the left-hand side is less or equal to a positive constant on the right-hand side. For this special type, we use a corresponding theorem of B.N. Pshenichny on establishing a penalty parameter for convex programs. For differently sized linear programs of the special type, we demonstrate that suitable penalty parameters can be computed by a procedure in GNU Octave based on GLPK software.

Driver Positioning and Incentive Budgeting with an Escrow Mechanism for Ride-Sharing Platforms

Drivers on the Lyft ride-share platform do not always know where the areas of supply shortage are in real time. This lack of information hurts both riders trying to find a ride and drivers trying to determine how to maximize their earnings opportunities. Lyft’s Personal Power Zone (PPZ) product helps the company to maintain high levels of service on the platform by influencing the spatial distribution of drivers in real time via monetary incentives that encourage them to reposition their vehicles. The underlying system that powers the product has two main components: (1) a novel “escrow mechanism” that tracks available incentive budgets tied to locations within a city in real time, and (2) an algorithm that solves the stochastic driver-positioning problem to maximize short-run revenue from riders’ fares. The optimization problem is a multiagent dynamic program that is too complicated to solve optimally for our large-scale application. Our approach is to decompose it into two subproblems. The first determines the set of drivers to incentivize and where to incentivize them to position themselves. The second determines how to fund each incentive using the escrow budget. By formulating it as two convex programs, we are able to use commercial solvers that find the optimal solution in a matter of seconds. Rolled out to all 320 cities in which Lyft operates in a little more than a year, the system now generates millions of bonuses that incentivize hundreds of thousands of active drivers to optimally position themselves in anticipation of ride requests every week. Together, the PPZ product and its underlying algorithms represent a paradigm shift in how Lyft drivers drive and generate earnings on the platform. Its direct business impact has been a 0.5% increase in incremental bookings, amounting to tens of millions of dollars per year. In addition, the product has brought about significant improvements to the driver and rider experience on the platform. These include statistically significant reductions in pick-up times and ride cancellations. Finally, internal surveys reveal that the vast majority of drivers prefer PPZs over the legacy system.

First-Order Methods for Constrained Convex Programming Based on Linearized Augmented Lagrangian Function

First-order methods (FOMs) have been popularly used for solving large-scale problems. However, many existing works only consider unconstrained problems or those with simple constraint. In this paper, we develop two FOMs for constrained convex programs, where the constraint set is represented by affine equations and smooth nonlinear inequalities. Both methods are based on the classical augmented Lagrangian function. They update the multipliers in the same way as the augmented Lagrangian method (ALM) but use different primal updates. The first method, at each iteration, performs a single proximal gradient step to the primal variable, and the second method is a block update version of the first one. For the first method, we establish its global iterate convergence and global sublinear and local linear convergence, and for the second method, we show a global sublinear convergence result in expectation. Numerical experiments are carried out on the basis pursuit denoising, convex quadratically constrained quadratic programs, and the Neyman-Pearson classification problem to show the empirical performance of the proposed methods. Their numerical behaviors closely match the established theoretical results.