Xuan Vinh Doan We propose a convex optimization formulation with the Ky Fan 2-k-norm and l1-norm to fi nd k largest approximately rank-one submatrix blocks of a given nonnegative matrix that has low-rank block diagonal structure with noise. We analyze low-rank and sparsity structures of the optimal solutions using properties of these two matrix norms. We show that, under … Read more
Cooperative games with transferable utilities belong to a branch of game theory where groups of players can enter into binding agreements and form coalitions in order to jointly achieve some objectives. In a cooperative setting, one of the most important questions to address is how to establish a payoff distribution among the players in such … Read more
We generalize the notions of user equilibrium and system optimum to non-atomic congestion games with stochastic demands. We establish upper bounds on the price of anarchy for three different settings of link cost functions and demand distributions, namely, (a) affine cost functions and general distributions, (b) polynomial cost functions and general positive-valued distributions, and (c) … Read more
In this paper, we develop a distributionally robust portfolio optimization model where the robustness is to different dependency structures among the random losses. For a Frechet class of distributions with overlapping marginals, we show that the distributionally robust portfolio optimization problem is efficiently solvable with linear programming. To guarantee the existence of a joint multivariate … Read more
We propose a proximal point algorithm to solve LAROS problem, that is the problem of finding a “large approximately rank-one submatrix”. This LAROS problem is used to sequentially extract features in data. We also develop a new stopping criterion for the proximal point algorithm, which is based on the duality conditions of \eps-optimal solutions of … Read more
Current successful methods for solving semidefinite programs, SDP, are based on primal-dual interior-point approaches. These usually involve a symmetrization step to allow for application of Newton’s method followed by block elimination to reduce the size of the Newton equation. Both these steps create ill-conditioning in the Newton equation and singularity of the Jacobian of the … Read more
We propose a convex optimization formulation with the nuclear norm and $\ell_1$-norm to find a large approximately rank-one submatrix of a given nonnegative matrix. We develop optimality conditions for the formulation and characterize the properties of the optimal solutions. We establish conditions under which the optimal solution of the convex formulation has a specific sparse … Read more
Given a combinatorial optimization problem with an arbitrary partition of the set of random objective coefficients, we evaluate the tightest possible bound on the expected optimal value for joint distributions consistent with the given multivariate marginals of the subsets in the partition. For univariate marginals, this bound was first proposed by Meilijson and Nadas (Journal … Read more
In this paper, we propose a semidefinite optimization (SDP) based model for the class of minimax two-stage stochastic linear optimization problems with risk aversion. The distribution of the second-stage random variables is assumed to be chosen from a set of multivariate distributions with known mean and second moment matrix. For the minimax stochastic problem with … Read more
We propose a moment relaxation for two problems, the separation and covering problem with semi-algebraic sets generated by a polynomial of degree d. We show that (a) the optimal value of the relaxation finitely converges to the optimal value of the original problem, when the moment order r increases and (b) there exist probability measures … Read more