  


Nonlinear optimization for matroid intersection and extensions
Yael Berstein(yaelbertx.technion.ac.il) Abstract: We address optimization of nonlinear functions of the form $f(Wx)$~, where $f:\R^d\rightarrow \R$ is a nonlinear function, $W$ is a $d\times n$ matrix, and feasible $x$ are in some large finite set $\calF$ of integer points in $\R^n$~. Generally, such problems are intractable, so we obtain positive algorithmic results by looking at broad natural classes of $f$~, $W$ and $\calF$~. One of our main motivations is multiobjective discrete optimization, where $f$ trades off the linear functions given by the rows of $W$~. Another motivation is that we want to extend as much as possible the known results about polynomialtime linear optimization over trees, assignments, matroids, polymatroids, etc. to nonlinear optimization over such structures. We assume that the convex hull of $\calF$ is welldescribed by linear inequalities (i.e., we have an efficient separation oracle). For example, the set of characteristic vectors of common bases of a pair of matroids on a common ground set satisfies this property for $\calF$~. In this setting, the problem is already known to be intractable (even for a single matroid), for general $f$ (given by a comparison oracle), for (i) $d=1$ and binaryencoded $W$~, and for (ii) $d=n$ and $W=I$~. Our main results (a few technicalities suppressed): 1 When $\calF$ is well described, $f$ is convex (or even quasiconvex), and $W$ has a fixed number of rows and is unary encoded or with entries in a fixed set, we give an efficient deterministic algorithm for maximization. 2 When $\calF$ is well described, $f$ is a norm, and binaryencoded $W$ is nonnegative, we give an efficient deterministic constantapproximation algorithm for maximization. 3 When $\calF$ is well described, $f$ is ``ray concave'' and nondecreasing, and $W$ has a fixed number of rows and is unary encoded or with entries in a fixed set, we give an efficient deterministic constantapproximation algorithm for minimization. 4 When $\calF$ is the set of characteristic vectors of common bases of a pair of vectorial matroids on a common ground set, $f$ is arbitrary, and $W$ has a fixed number of rows and is unary encoded, we give an efficient randomized algorithm for optimization. Keywords: matroid intersection, welldescribed polytope, nonlinear optimization Category 1: Combinatorial Optimization Category 2: Nonlinear Optimization Citation: Download: [PDF] Entry Submitted: 07/22/2008 Modify/Update this entry  
Visitors  Authors  More about us  Links  
Subscribe, Unsubscribe Digest Archive Search, Browse the Repository

Submit Update Policies 
Coordinator's Board Classification Scheme Credits Give us feedback 
Optimization Journals, Sites, Societies  