197 research outputs found
A partial differential equation for the strictly quasiconvex envelope
In a series of papers Barron, Goebel, and Jensen studied Partial Differential
Equations (PDE)s for quasiconvex (QC) functions \cite{barron2012functions,
barron2012quasiconvex,barron2013quasiconvex,barron2013uniqueness}. To overcome
the lack of uniqueness for the QC PDE, they introduced a regularization: a PDE
for \e-robust QC functions, which is well-posed. Building on this work, we
introduce a stronger regularization which is amenable to numerical
approximation. We build convergent finite difference approximations, comparing
the QC envelope and the two regularization. Solutions of this PDE are strictly
convex, and smoother than the robust-QC functions.Comment: 20 pages, 6 figures, 1 tabl
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
On Unconstrained Quasi-Submodular Function Optimization
With the extensive application of submodularity, its generalizations are
constantly being proposed. However, most of them are tailored for special
problems. In this paper, we focus on quasi-submodularity, a universal
generalization, which satisfies weaker properties than submodularity but still
enjoys favorable performance in optimization. Similar to the diminishing return
property of submodularity, we first define a corresponding property called the
{\em single sub-crossing}, then we propose two algorithms for unconstrained
quasi-submodular function minimization and maximization, respectively. The
proposed algorithms return the reduced lattices in iterations,
and guarantee the objective function values are strictly monotonically
increased or decreased after each iteration. Moreover, any local and global
optima are definitely contained in the reduced lattices. Experimental results
verify the effectiveness and efficiency of the proposed algorithms on lattice
reduction.Comment: 11 page
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