A Scalable Algorithm For Sparse Portfolio Selection

The sparse portfolio selection problem is one of the most famous and frequently-studied problems in the optimization and financial economics literatures. In a universe of risky assets, the goal is to construct a portfolio with maximal expected return and minimum variance, subject to an upper bound on the number of positions, linear inequalities and minimum investment constraints. Existing certifiably optimal approaches to this problem do not converge within a practical amount of time at real world problem sizes with more than 400 securities. In this paper, we propose a more scalable approach. By imposing a ridge regularization term, we reformulate the problem as a convex binary optimization problem, which is solvable via an efficient outer-approximation procedure. We propose various techniques for improving the performance of the procedure, including a heuristic which supplies high-quality warm-starts, a preprocessing technique for decreasing the gap at the root node, and an analytic technique for strengthening our cuts. We also study the problem's Boolean relaxation, establish that it is second-order-cone representable, and supply a sufficient condition for its tightness. In numerical experiments, we establish that the outer-approximation procedure gives rise to dramatic speedups for sparse portfolio selection problems.Comment: Submitted to INFORMS Journal on Computin

Towards Effective Exact Algorithms for the Maximum Balanced Biclique Problem

The Maximum Balanced Biclique Problem (MBBP) is a prominent model with numerous applications. Yet, the problem is NP-hard and thus computationally challenging. We propose novel ideas for designing effective exact algorithms for MBBP. Firstly, we introduce an Upper Bound Propagation procedure to pre-compute an upper bound involving each vertex. Then we extend an existing branch-and-bound algorithm by integrating the pre-computed upper bounds. We also present a set of new valid inequalities induced from the upper bounds to tighten an existing mathematical formulation for MBBP. Lastly, we investigate another exact algorithm scheme which enumerates a subset of balanced bicliques based on our upper bounds. Experiments show that compared to existing approaches, the proposed algorithms and formulations are more efficient in solving a set of random graphs and large real-life instances

Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms

We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a $d$-degenerate graph $G$ and an integer $k$, outputs an independent set $Y$, such that for every independent set $X$ in $G$ of size at most $k$, the probability that $X$ is a subset of $Y$ is at least $\left({(d+1)k \choose k} \cdot k(d+1)\right)^{-1}$.The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph $G$, a set $T = \{\{s_1, t_1\}, \{s_2, t_2\}, \ldots, \{s_\ell, t_\ell\}\}$ of terminal pairs and an integer $k$, returns an induced subgraph $G^\star$ of $G$ that maintains all the inclusion minimal multicuts of $G$ of size at most $k$, and does not contain any $(k+2)$-vertex connected set of size $2^{{\cal O}(k)}$. In particular, $G^\star$ excludes a clique of size $2^{{\cal O}(k)}$ as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for Stable $s$-$t$ Separator, Stable Odd Cycle Transversal and Stable Multicut on general graphs, and for Stable Directed Feedback Vertex Set on $d$-degenerate graphs, resolving two problems left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our algorithms can be derandomized at the cost of a small overhead in the running time.Comment: 35 page

A hybrid constraint programming and semidefinite programming approach for the stable set problem

This work presents a hybrid approach to solve the maximum stable set problem, using constraint and semidefinite programming. The approach consists of two steps: subproblem generation and subproblem solution. First we rank the variable domain values, based on the solution of a semidefinite relaxation. Using this ranking, we generate the most promising subproblems first, by exploring a search tree using a limited discrepancy strategy. Then the subproblems are being solved using a constraint programming solver. To strengthen the semidefinite relaxation, we propose to infer additional constraints from the discrepancy structure. Computational results show that the semidefinite relaxation is very informative, since solutions of good quality are found in the first subproblems, or optimality is proven immediately.Comment: 14 page