# Online Covering with Convex Objectives and Applications

@article{Azar2014OnlineCW, title={Online Covering with Convex Objectives and Applications}, author={Yossi Azar and Ilan Reuven Cohen and Debmalya Panigrahi}, journal={ArXiv}, year={2014}, volume={abs/1412.3507} }

We give an algorithmic framework for minimizing general convex objectives (that are differentiable and monotone non-decreasing) over a set of covering constraints that arrive online. This substantially extends previous work on online covering for linear objectives (Alon {\em et al.}, STOC 2003) and online covering with offline packing constraints (Azar {\em et al.}, SODA 2013). To the best of our knowledge, this is the first result in online optimization for generic non-linear objectives…

## 16 Citations

### Covering with Sum of l q-Norm Objectives

- Computer Science
- 2017

Improved online algorithms for non-uniform buy-at-bulk network design and the first online algorithm for throughput maximization under lp-norm edge capacities are obtained.

### Worst Case Competitive Analysis of Online Algorithms for Conic Optimization

- Computer Science, MathematicsArXiv
- 2016

This work derives a sufficient condition on the objective function that guarantees a constant worst case competitive ratio (greater than or equal to $\frac{1}{2}$) for monotone objective functions, and hence design effective smoothing customized for a given cost function.

### Designing smoothing functions for improved worst-case competitive ratio in online optimization

- Computer Science, MathematicsNIPS
- 2016

The worst case competitive ratio of two primal-dual algorithms for a class of online convex (conic) optimization problems is analyzed and it is shown that the optimal smoothing can be derived by solving a convex optimization problem.

### Online Convex Covering and Packing Problems

- Mathematics, Computer ScienceArXiv
- 2015

For any convex polynomial cost functions with non-negative coefficients and maximum degree $\tau$, an O(\tau \log n)^\tau$-competitive online convex covering algorithm, and an O(tau)-competitiveOnline convex packing algorithm are introduced, matching the known $\Omega(\Tau)$ and $Omega(n)$ lower bounds respectively.

### Online Packing and Covering Framework with Convex Objectives

- MathematicsArXiv
- 2014

Using this fractional solver with problem-dependent randomized rounding procedures, competitive algorithms are obtained for the following problems: online covering LPs minimizing $\ell_p$-norms of arbitrary packing constraints, set cover with multiple cost functions, capacity constrained facility location, capacitated multicast problem, setCover with set requests, and profit maximization with non-separable production costs.

### Online Algorithms for Covering and Packing Problems with Convex Objectives

- Computer Science2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
- 2016

A primal-dual approach is used to give online algorithms for covering and packing problems with (non-linear) convex objectives, and these algorithms are used to simplify, unify, and improve upon previous results for several applications.

### D S ] 1 4 A pr 2 01 5 Online Convex Covering and Packing Problems

- Mathematics
- 2018

We study the online convex covering problem and online convex packing problem. The (offline) convex covering problem is modeled by the following convex program: min ~ x∈Rn+ f(~x) s.t. A~x ≥ ~1 where…

### Online and Random-order Load Balancing Simultaneously

- Computer Science, MathematicsSODA
- 2017

This paper provides algorithms with simultaneous guarantees for the worst-case model as well as for the random-order model, where an arbitrary set of jobs comes in random order, and proposes algorithm SimultaneousLB, a new algorithm with improved regret for Online Linear Optimization over the non-negative vectors in the lq ball.

### Online covering with $$\ell _q$$ℓq-norm objectives and applications to network design

- Computer ScienceMath. Program.
- 2020

An improved online algorithm for non-uniform buy-at-bulk network design and a poly-logarithmic competitive ratio for throughput maximization under ℓp-norm capacities are obtained.

### Online Covering with Sum of $\ell_q$-Norm Objectives

- Computer ScienceICALP
- 2017

Improved online algorithms for non-uniform buy-at-bulk network design and the first online algorithm for throughput maximization under $\ell_p$-norm edge capacities are obtained.

## References

SHOWING 1-10 OF 36 REFERENCES

### Online Mixed Packing and Covering

- Computer Science, MathematicsSODA
- 2013

This work demonstrates that the classical framework of solving optimization problems by obtaining a fractional solution to a linear program and rounding it to an integer solution can be extended to the online setting using primal-dual techniques and obtains the first algorithm that obtains a polylogarithmic competitive ratio for solving mixed LPs online.

### Online Primal-Dual Algorithms for Covering and Packing

- Computer Science, MathematicsMath. Oper. Res.
- 2009

This work provides general deterministic primal-dual algorithms for online fractional covering and packing problems and also provides deterministic algorithms for several integral online covering andpacking problems.

### Approximation algorithms for scheduling on multiple machines

- Computer Science46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
- 2005

We develop a single rounding algorithm for scheduling on unrelated parallel machines; this algorithm works well with the known linear programming, quadratic programming, and convex…

### Convex programming for scheduling unrelated parallel machines

- Computer Science, MathematicsSTOC '05
- 2005

This work considers the classical problem of scheduling parallel unrelated machines and provides a 2-approximation algorithm for any fixed lp norm (p>1) and gives the first time that general convex programming techniques are used in the area of scheduling.

### Better bounds for online load balancing on unrelated machines

- Computer ScienceSODA '08
- 2008

The first randomized online algorithms which outperform deterministic ones under any (integral) Lp norm for p = 2,…,137 are presented, providing a positive answer to the question whether randomization helps online load balancing under Lp norms on unrelated machines.

### Online Node-Weighted Steiner Forest and Extensions via Disk Paintings

- Mathematics, Computer Science2013 IEEE 54th Annual Symposium on Foundations of Computer Science
- 2013

The central idea in this technique is to amortize the cost of primal updates to a set of carefully selected mutually disjoint fixed-radius dual disks centered at a subset of terminals to form a new framework for online network design problems that is called disk paintings.

### Online Primal-Dual for Non-linear Optimization with Applications to Speed Scaling

- Computer ScienceWAOA
- 2012

This analysis shows that competitive algorithms exist for problems that had resisted analysis using the dominant potential function approach in the speed-scaling literature, and provides alternate cleaner analysis for other known results.

### Towards the randomized k-server conjecture: a primal-dual approach

- Computer Science, MathematicsSODA '10
- 2010

Recently, Coté et al. [10] proposed an approach for solving the k-server problem on Hierchically Separated Trees (HSTs). In particular, they define a problem on a uniform metric, and show that if an…

### The competitiveness of on-line assignments

- Mathematics, Computer ScienceSODA '92
- 1992

It is concluded that for the on-line problem where a number of servers are ready to provide service to a set of customers, randomized algorithms differ from deterministic ones by precisely a constant factor.

### Generalized machine activation problems

- Business, Computer ScienceSODA '11
- 2011

A greedy algorithm is developed that yields a fractional assignment of jobs, such that at least n − ε jobs are assigned fractionally and the total cost is at most 1 + ln(n/ε) times the optimum, which gives an affirmative answer to the open question posed in earlier work on universal facility location.