We present a unified semidefinite programming hierarchies rounding approximation algorithm for a class of maximum graph bisection problems with improved approximation ratios. Under the above algorithmic framework, we ...
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We present a unified semidefinite programming hierarchies rounding approximation algorithm for a class of maximum graph bisection problems with improved approximation ratios. Under the above algorithmic framework, we show that the approximation ratios of MAX-n/2-CUT, MAX-n/2-DENSE-SUBGRAPH, and MAX-n/ 2-VERTEX-COVER are equal to those of MAX-n/2-UNCUT, MAX-n/2-DIRECTED-CUT, and MAX-n/2-DIRECTED-UNCUT, respectively.
This paper studies the problem of deterministic rank-one matrix completion. It is known that the simplest semidefiniteprogramming relaxation, involving minimization of the nuclear norm, does not in general return the...
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This paper studies the problem of deterministic rank-one matrix completion. It is known that the simplest semidefiniteprogramming relaxation, involving minimization of the nuclear norm, does not in general return the solution for this problem. In this paper, we show that in every instance where the problem has a unique solution, one can provably recover the original matrix through the level 2 Lasserre relaxation with minimization of the trace norm. We further show that the solution of the proposed semidefinite program is Lipschitz stable with respect to perturbations of the observed entries, unlike more basic algorithms such as nonlinear propagation or ridge regression. Our proof is based on recursively building a certificate of optimality corresponding to a dual sum-of-squares (SoS) polynomial. This SoS polynomial is built from the polynomial ideal generated by the completion constraints and the monomials provided by the minimization of the trace. The proposed relaxation fits in the framework of the Lasserre hierarchy, albeit with the key addition of the trace objective function. Finally, we show how to represent and manipulate the moment tensor in favorable complexity by means of a hierarchical low-rank factorization.
We consider the generalized moment problem (GMP) over the simplex and the sphere. This is a rich setting and it contains NP-hard problems as special cases, like constructing optimal cubature schemes and rational optim...
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We consider the generalized moment problem (GMP) over the simplex and the sphere. This is a rich setting and it contains NP-hard problems as special cases, like constructing optimal cubature schemes and rational optimization. Using the reformulation-linearization technique (RLT) and Lasserre-type hierarchies, relaxations of the problem are introduced and analyzed. For our analysis we assume throughout the existence of a dual optimal solution as well as strong duality. For the GMP over the simplex we prove a convergence rate of O(1/r) for a linear programming, RLT-type hierarchy, where r is the level of the hierarchy, using a quantitative version of Polya's Positivstellensatz. As an extension of a recent result by Fang and Fawzi (Math Program, 2020. https://***/10.1007/s10107-020-01537-7) we prove the Lasserre hierarchy of the GMP (Lasserre in Math Program 112(1):65-92, 2008. https://***/10.1007/s10107-006-(X)85-1) over the sphere has a convergence rate of O(1/r(2)). Moreover, we show the introduced linear RLT-relaxation is a generalization of a hierarchy for minimizing forms of degree d over the simplex, introduced by De Klerk et al. (J Theor Comput Sci 361(2-3):210-225, 2006).
In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper-and the lower-level problems are polynomials. We present methods for finding its ...
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In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper-and the lower-level problems are polynomials. We present methods for finding its global minimizers and global minimum using a sequence of semidefiniteprogramming (SDP) relaxations and provide convergence results for the methods. Our scheme for problems with a convex lower-level problem involves solving a transformed equivalent single-level problem by a sequence of SDP relaxations, whereas our approach for general problems involving a nonconvex polynomial lower-level problem solves a sequence of approximation problems via another sequence of SDP relaxations.
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