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Strong Duality, Complementarity, and Duality Gaps\\
in Conic Convex Optimization
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     {Simon Schurr}$^a$, {Levent Tun{\c{c}}el}$^a$, and Henry
Wolkowicz$^a$\\[14pt]

     $^a${University of Waterloo}\\
%\href{http://www.math.uwaterloo.ca/CandO_Dept/homepage.html}
{Dept. of Combinatorics \& Optimization}\\
          Waterloo, Ontario  Canada\\[3mm]

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For convex minimization problems with
optimal value $v_P$, {\em weak duality} 
holds in general, i.e.~the optimal value of 
the Lagrangian dual provides a lower bound,
$v_D \leq v_P$. However, 
{\em strong duality} (i.e.~$v_D = v_P$ and $v_D$ is attained)
requires a {\em constraint qualification, CQ}, e.g. the Slater (strict
feasibility) CQ.
In the absence of a CQ, the dual optimal value $v_D$ may be unattained
and/or there can be a positive duality gap $v_P - v_D>0$.
In addition, the (near) absence of a CQ results in numerical difficulties.

 We first present known and new
conditions that guarantee strong duality without any
constraint qualification. These are based on {\em minimum representations}:
known results using the {\em minimal face} 
and new results using the {\em minimal subspace}.

 We then study conditions that guarantee a positive duality gap.
Necessary and sufficient conditions for a positive duality gap
are given, and it is shown how these can be used to generate instances
satisfying this property.
We then present an algorithm that solves in an efficient and stable
way, feasible conic convex optimization problems, including those
for which the Slater constraint qualification fails.
In addition, we present new relations between strict
complementarity and duality gaps.

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