We study the computational complexity of the infinite-horizon discounted-reward Markov Decision Problem (MDP) with a finite state space $\cS$ and a finite action space $\cA$. We show that any randomized algorithm needs a running time at least $\Omega(\carS^2\carA)$ to compute an $\epsilon$-optimal policy with high probability. We consider two variants of the MDP where the … Read more
We propose a randomized linear programming algorithm for approximating the optimal policy of the discounted Markov decision problem. By leveraging the value-policy duality, the algorithm adaptively samples state transitions and makes exponentiated primal-dual updates. We show that it finds an ε-optimal policy using nearly-linear running time in the worst case. For Markov decision processes that … Read more
The algorithm proposed in this paper runs in a polynomial oracle time, i.e., in a number of arithmetic operations and calls to the separation oracle bounded by a polynomial in the number of variables and in the maximum binary size of an entry of the coefficient matrix. This algorithm is much simpler than traditional polynomial … Read more
The basis matrices corresponding to consecutive iterations of the simplex method only differ in a single column. This fact is commonly exploited in current LP solvers to avoid having to compute a new factorization of the basis at every iteration. Instead, a previous factorization is updated to reflect the modified column. Several methods are known … Read more
Radiation therapy is considered to be one of important treatment protocols for cancers. Radiation therapy employs several beams of ionizing radiation to kill cancer tumors, but such irradiation also causes damage to normal tissues. Therefore, a treatment plan should satisfy dose-volume constraints (DVCs). Intensity-modulated radiotherapy treatment (IMRT) enables to control the beam intensities and gives … Read more
We propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix $A\in \R^{m\times n}$, the {\em kernel problem} requires a positive vector in the kernel of $A$, and the {\em image problem} requires a positive vector in the image of $A^\T$. Both algorithms iterate between simple first order steps and rescaling steps. … Read more
We deal with a recently proposed method of Chubanov [1] for solving linear homogeneous systems with positive variables. Some improvements of Chubanov’s method and its analysis are presented. We propose a new and simple cut criterion and show that the cuts defined by the new criterion are at least as sharp as in [1]. The … Read more
The analysis of infeasible subproblems plays an import role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems. The first is to analyze the sequence of implications obtained by domain propagation that led to infeasibility. The … Read more
We show how random projections can be used to solve large-scale dense linear programs approximately. This is a new application of techniques which are now fairly well known in probabilistic algorithms, but have never yet been systematically applied to the fundamental class of Linear Programming. We develop the necessary theoretical framework, and show that this … Read more
In this paper we propose a primal-dual homotopy method for $\ell_1$-minimization problems with infinity norm constraints in the context of sparse reconstruction. The natural homotopy parameter is the value of the bound for the constraints and we show that there exists a piecewise linear solution path with finitely many break points for the primal problem … Read more