arXiv subjects

Nicholas Moehle

Publications and source records attributed to Nicholas Moehle.

11 recordsLinked to original sources

arXiv2023

Optimal Racing of an Energy-Limited Vehicle

We consider the problem of controlling a vehicle to arrive at a fixed destination while minimizing a combination of energy consumption and travel time. Our model includes vehicle speed and accelaration limits, aerodynamic drag, rolling resistance, nonlinear engine losses, and internal energy limits. The naive problem formulation is not convex; however, we show that a simple convex relaxation is tight. We provide a numerical example, and discuss extensions to vehicles with unconventional drivetrains, such as hybrid vehicles and solar cars.

math.OC↗

arXiv2022

Portfolio Construction as Linearly Constrained Separable Optimization

Mean-variance portfolio optimization problems often involve separable nonconvex terms, including penalties on capital gains, integer share constraints, and minimum position and trade sizes. We propose a heuristic algorithm for such problems based on the alternating direction method of multipliers (ADMM). This method allows for solve times in tens to hundreds of milliseconds with around 1000 securities and 100 risk factors. We also obtain a bound on the achievable performance. Our heuristic and bound are both derived from similar results for other optimization problems with a separable objective and affine equality constraints. We discuss a concrete implementation in the case where the separable terms in the objective are piecewise quadratic, and we empirically demonstrate its effectiveness for tax-aware portfolio construction.

math.OC↗

arXiv2022

Risk-Sensitive Model Predictive Control

We present a heuristic policy and performance bound for risk-sensitive convex stochastic control that generalizes linear-exponential-quadratic regulator (LEQR) theory. Our heuristic policy extends standard, risk-neutral model predictive control (MPC); however, instead of ignoring uncertain noise terms, our policy assumes these noise terms turns out either favorably or unfavorably, depending on a risk aversion parameter. In the risk-seeking case, this modified planning problem is convex. In the risk-averse case, it requires minimizing a difference of convex functions, which is done (approximately) using the convex-concave procedure. In both cases, we obtain a lower bound on the optimal cost as a by-product of solving the planning problem. We give a numerical example of controlling a battery to power an uncertain load, and show that our policy reduces the risk of a very bad outcome (as compared with standard certainty equivalent control) with negligible impact on the the average performance.

math.OC↗

arXiv2021

Tax-Aware Portfolio Construction via Convex Optimization

We describe an optimization-based tax-aware portfolio construction method that adds tax liability to standard Markowitz-based portfolio construction. Our method produces a trade list that specifies the number of shares to buy of each asset and the number of shares to sell from each tax lot held. To avoid wash sales (in which some realized capital losses are disallowed), we assume that we trade monthly, and cannot simultaneously buy and sell the same asset. The tax-aware portfolio construction problem is not convex, but it becomes convex when we specify, for each asset, whether we buy or sell it. It can be solved using standard mixed-integer convex optimization methods at the cost of very long solve times for some problem instances. We present a custom convex relaxation of the problem that borrows curvature from the risk model. This relaxation can provide a good approximation of the true tax liability, while greatly enhancing computational tractability. This method requires the solution of only two convex optimization problems: the first determines whether we buy or sell each asset, and the second generates the final trade list. In our numerical experiments, our method almost always solves the nonconvex problem to optimality, and when it does not, it produces a trade list very close to optimal. Backtests show that the performance of our method is indistinguishable from that obtained using a globally optimal solution, but with significantly reduced computational effort.

math.OC↗

arXiv2021

Portfolio Performance Attribution via Shapley Value

We consider an investment process that includes a number of features, each of which can be active or inactive. Our goal is to attribute or decompose an achieved performance to each of these features, plus a baseline value. There are many ways to do this, which lead to potentially different attributions in any specific case. We argue that a specific attribution method due to Shapley is the preferred method, and discuss methods that can be used to compute this attribution exactly, or when that is not practical, approximately.

q-fin.CP↗

arXiv2021

A Certainty Equivalent Merton Problem

The Merton problem is the well-known stochastic control problem of choosing consumption over time, as well as an investment mix, to maximize expected constant relative risk aversion (CRRA) utility of consumption. Merton formulated the problem and provided an analytical solution in 1970; since then a number of extensions of the original formulation have been solved. In this note we identify a certainty equivalent problem, i.e., a deterministic optimal control problem with the same optimal value function and optimal policy, for the base Merton problem, as well as a number of extensions. When time is discretized, the certainty equivalent problem becomes a second-order cone program (SOCP), readily formulated and solved using domain specific languages for convex optimization. This makes it a good starting point for model predictive control, a policy that can handle extensions that are either too cumbersome or impossible to handle exactly using standard dynamic programming methods.

math.OC↗

arXiv2019

Dynamic Energy Management

We present a unified method, based on convex optimization, for managing the power produced and consumed by a network of devices over time. We start with the simple setting of optimizing power flows in a static network, and then proceed to the case of optimizing dynamic power flows, i.e., power flows that change with time over a horizon. We leverage this to develop a real-time control strategy, model predictive control, which at each time step solves a dynamic power flow optimization problem, using forecasts of future quantities such as demands, capacities, or prices, to choose the current power flow values. Finally, we consider a useful extension of model predictive control that explicitly accounts for uncertainty in the forecasts. We mirror our framework with an object-oriented software implementation, an open-source Python library for planning and controlling power flows at any scale. We demonstrate our method with various examples. Appendices give more detail about the package, and describe some basic but very effective methods for constructing forecasts from historical data.

math.OC↗

arXiv2017

A Distributed Method for Optimal Capacity Reservation

We consider the problem of reserving link capacity in a network in such a way that any of a given set of flow scenarios can be supported. In the optimal capacity reservation problem, we choose the reserved link capacities to minimize the reservation cost. This problem reduces to a large linear program, with the number of variables and constraints on the order of the number of links times the number of scenarios. Small and medium size problems are within the capabilities of generic linear program solvers. We develop a more scalable, distributed algorithm for the problem that alternates between solving (in parallel) one flow problem per scenario, and coordination steps, which connect the individual flows and the reservation capacities.

cs.DC↗

arXiv2016

Value Function Approximation for Direct Control of Switched Power Converters

We consider the problem of controlling switched-mode power converters using model predictive control. Model predictive control requires solving optimization problems in real time, limiting its application to systems with small numbers of switches and a short horizon. We propose a technique for using off-line computation to approximate the model predictive controller. This is done by dividing the planning horizon into two segments, and using a quadratic function to approximate the optimal cost over the second segment. The approximate model predictive algorithm minimizes the true cost over the first segment, and the approximate cost over the second segment, allowing the user to adjust the computational requirements by changing the length of the first segment. We conclude with two simulated examples.

eess.SY↗

arXiv2016

Optimal Current Waveforms for Switched-Reluctance Motors

In this paper, we address the problem of finding current waveforms for a switched reluctance motor that minimize a user-defined combination of torque ripple and RMS current. The motor model we use is fairly general, and includes magnetic saturation, voltage and current limits, and highly coupled magnetics (and therefore, unconventional geometries and winding patterns). We solve this problem by approximating it as a mixed-integer convex program, which we solve globally using branch and bound. We demonstrate our approach on an experimentally verified model of a fully pitched switched reluctance motor, for which we find the globally optimal waveforms, even for high rotor speeds.

eess.SY↗

arXiv2015

A Simple Effective Heuristic for Embedded Mixed-Integer Quadratic Programming

In this paper we propose a fast optimization algorithm for approximately minimizing convex quadratic functions over the intersection of affine and separable constraints (i.e., the Cartesian product of possibly nonconvex real sets). This problem class contains many NP-hard problems such as mixed-integer quadratic programming. Our heuristic is based on a variation of the alternating direction method of multipliers (ADMM), an algorithm for solving convex optimization problems. We discuss the favorable computational aspects of our algorithm, which allow it to run quickly even on very modest computational platforms such as embedded processors. We give several examples for which an approximate solution should be found very quickly, such as management of a hybrid-electric vehicle drivetrain and control of switched-mode power converters. Our numerical experiments suggest that our method is very effective in finding a feasible point with small objective value; indeed, we find that in many cases, it finds the global solution.

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