[Paper Review] Online Covering with Convex Objectives and Applications

Yossi Azar, Ilan Reuven Cohen+1 more|arXiv (Cornell University)|Dec 11, 2014

Optimization and Search Problems20 references9 citations

TL;DR

This paper introduces a novel online algorithmic framework for minimizing general convex, differentiable, and non-decreasing objectives under online covering constraints, extending prior work on linear objectives. It applies this framework to the unrelated machine scheduling problem with startup costs and ℓp norms, achieving nearly optimal competitive ratios via a two-phase approach of fractional optimization followed by tailored online rounding, with improved bounds for the ℓ1 norm case.

ABSTRACT

Motivated by the importance of energy storage networks in smart grids, we provide an algorithmic study of the online energy storage management problem in a network setting, the first to the best of our knowledge. Given online power supplies, either entirely renewable supplies or those in combination with traditional supplies, we want to route power from the supplies to demands using storage units subject to a decay factor. Our goal is to maximize the total utility of satisfied demands less the total production cost of routed power. We model renewable supplies with the zero production cost function and traditional supplies with convex production cost functions. For two natural storage unit settings, private and public, we design poly-logarithmic competitive algorithms in the network flow model using the dual fitting and online primal dual methods for convex problems. Furthermore, we show strong hardness results for more general settings of the problem. Our techniques may be of independent interest in other routing and storage management problems.

Motivation & Objective

• To develop a general online algorithmic framework for minimizing convex, differentiable, and non-decreasing objectives under online covering constraints.
• To extend existing online covering frameworks—previously limited to linear objectives or offline packing constraints—to handle general non-linear convex objectives.
• To apply the framework to the unrelated machine scheduling problem with startup costs and ℓp norms, including the challenging ℓ1 norm (total load).
• To design a competitive online rounding algorithm for the ℓ1 norm case that achieves a better competitive ratio than the generic ℓp rounding approach.
• To prove asymptotic tightness of the competitive ratios for both the general ℓp and specialized ℓ1 cases.

Proposed method

• Proposes a deterministic online algorithm for minimizing convex objectives subject to online covering constraints using a fractional solution phase followed by online rounding.
• Adapts ideas from online fractional optimization and offline rounding techniques for non-linear objectives to handle non-positive entries in the constraint matrix.
• Introduces a two-phase approach: first compute a competitive fractional solution using insights from the general convex framework, then round it online using a machine-copying technique with blue and red copies.
• For ℓ1 norm, designs a specialized rounding rule that assigns jobs based on a minimal prefix of machines with high fractional coverage, reducing the probability of costly red-copy assignments.
• Uses conditional probability and stochastic dominance to bound expected increases in the ℓp norm and machine cost, leveraging integrals and concentration inequalities.
• Applies Theorem 25 to combine bounds on expected ℓp norm and cost, deriving competitive ratios in terms of Φ (the fractional objective value) and logarithmic factors.

Experimental results

Research questions

• RQ1 Can a general online framework be developed for minimizing convex, non-linear objectives under online covering constraints, beyond the known linear cases?
• RQ2 How can the two-phase approach—fractional solution followed by online rounding—be adapted when the objective is non-linear and the constraint matrix contains negative entries?
• RQ3 What competitive ratio can be achieved for the unrelated machine scheduling problem with startup costs and ℓp norms, particularly for the ℓ1 norm case?
• RQ4 Is there a way to improve the competitive ratio for the ℓ1 norm case beyond the generic ℓp rounding framework?
• RQ5 Are the derived competitive ratios asymptotically tight for both the general ℓp and specialized ℓ1 settings?

Key findings

• The paper achieves a competitive ratio of O((log m)^{1/p} log n) for the ℓp norm scheduling problem, which is tight up to a logarithmic factor.
• For the ℓ1 norm case, a specialized rounding algorithm achieves a competitive ratio of O(log n), which is better than the generic ℓp rounding and asymptotically tight.
• The expected cost of machines opened by the algorithm is bounded by O(log n)Φ, where Φ is the fractional objective value.
• The expected ℓ1 norm of machine loads is at most 2Φ, showing that the rounding preserves the fractional solution quality in expectation.
• The probability of assigning a job to a red copy (i.e., a machine opened after the job arrives) is at most 1/n, which is crucial for cost control.
• The framework generalizes prior work on online covering with linear objectives and mixed packing-covering constraints, providing a unified approach for non-linear objectives.