Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Robust Algorithms for Online Convex Problems via Primal-Dual   Primal-dual methods in online optimization give several of the state-of-the
art results in both of the most common models: adversarial and
stochastic/random order. Here we try to provide a more unified analysis of
primal-dual algorithms to better understand the mechanisms behind this
important method. With this we are able of recover and extend in one goal
several results of the literature.
  In particular, we obtain robust online algorithm for fairly general online
convex problems: we consider the MIXED model where in some of the time steps
the data is stochastic and in the others the data is adversarial. Both the
quantity and location of the adversarial time steps are unknown to the
algorithm. The guarantees of our algorithms interpolate between the (close to)
best guarantees for each of the pure models. In particular, the presence of
adversarial times does not degrade the guarantee relative to the stochastic
part of the instance.
  Concretely, we first consider Online Convex Programming: at each time a
feasible set $V_t$ is revealed, and the algorithm needs to select $v_t \in V_t$
to minimize the total cost $\psi(\sum_t v_t)$, for a convex function $\psi$.
Our robust primal-dual algorithm for this problem on the MIXED model recovers
and extends, for example, a result of Gupta et al. and recent work on
$\ell_p$-norm load balancing by the author. We also consider the problem of
Welfare Maximization with Convex Production Costs: at each time a customer
presents a value $c_t$ and resource consumption vector $a_t$, and the goal is
to fractionally select customers to maximize the profit $\sum_t c_t x_t -
\psi(\sum_t a_t x_t)$. Our robust primal-dual algorithm on the MIXED model
recovers and extends the result of Azar et al.
  Given the ubiquity of primal-dual algorithms we hope the ideas presented here
will be useful in obtaining other robust algorithm in the MIXED or related
models.

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The categories of this paper are: q-bio
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom