Given a feasible solution x 0 to a mixed-integer program (MIP), the inverse MIP problem is to find an objective d such that x 0 is optimal for the MIP with objective function d, and among all such objectives, the distance from a given target objective is minimized. By using a novel expression for the MIP value function, we formulate the inverse MIP problem as a linear program (LP), albeit of exponentially large size.
We consider the maximum likelihood estimation of sparse inverse covariance matrices. We demonstrate that current heuristic approaches primarily encourage robustness, instead of the desired sparsity.We give a novel approach that solves the cardinality constrained likelihood problem to certifiable optimality. The approach uses techniques from mixed-integer optimization and convex optimization, and provides a high-quality solution with a guarantee on its suboptimality, even if the algorithm is terminated early. Using a variety of synthetic and real datasets, we demonstrate that our approach can solve problems where the dimension of the inverse covariance matrix is up to 1, 000s. We also demonstrate that our approach produces significantly sparser solutions than Glasso and other popular learning procedures, makes less false discoveries, while still maintaining state-of-the-art accuracy.
The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of m linear inequalities in n variables (P ) : A ⊤ x ≤ u when the set of solutions P := {x ∈ R n : A ⊤ x ≤ u} has positive volume. However, when (P ) is infeasible, the ellipsoid algorithm has no built-in mechanism for proving that (P ) is infeasible. This is in contrast to the other two fundamental algorithms for tackling (P ), namely the simplex method and interior-point methods, each of which can be easily implemented in a way that either produces a solution of (P ) or proves that (P ) is infeasible by producing a solution to the alternative system (Alt) : Aλ = 0, u ⊤ λ < 0, λ ≥ 0. Motivated by this, we develop an Oblivious Ellipsoid Algorithm (OEA) that produces a solution of (P ) when (P ) is feasible and produces a solution of (Alt) when (P ) is infeasible, and whose operations complexity dependence on the dimensions (which we describe as dimension complexity below) in the regime m ≫ n is on the same order as a standard ellipsoid method (O(m 4 )) when (P ) is infeasible (but has worse dimension complexity when (P ) is feasible). However, if one is only interested in proving infeasibility and not necessarily producing a solution of (Alt), then we show that a simplified version of OEA achieves O(m 3 n) dimension complexity, which is superior to the O(m 4 ) dimension complexity bound of a standard ellipsoid algorithm applied to solve (Alt). Introduction, preliminaries, and summary of resultsGiven data (A, u) ∈ R n×m × R m , the ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of linear inequalities (P ) : A ⊤ x ≤ u when the set of solutions P := {x ∈ R n : A ⊤ x ≤ u} has positive volume. However, when (P ) is infeasible, existing versions of the ellipsoid algorithm have no mechanism for deciding if (P )
The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of m linear inequalities in n variables [Formula: see text] when its set of solutions has positive volume. However, when [Formula: see text] is infeasible, the ellipsoid algorithm has no mechanism for proving that (P) is infeasible. This is in contrast to the other two fundamental algorithms for tackling [Formula: see text], namely, the simplex and interior-point methods, each of which can be easily implemented in a way that either produces a solution of [Formula: see text] or proves that [Formula: see text] is infeasible by producing a solution to the alternative system [Formula: see text]. This paper develops an oblivious ellipsoid algorithm (OEA) that either produces a solution of [Formula: see text] or produces a solution of [Formula: see text]. Depending on the dimensions and other condition measures, the computational complexity of the basic OEA may be worse than, the same as, or better than that of the standard ellipsoid algorithm. We also present two modified versions of OEA, whose computational complexity is superior to that of OEA when [Formula: see text]. This is achieved in the first modified version by proving infeasibility without producing a solution of [Formula: see text], and in the second version by using more memory. Funding: J. Lamperski and R. M. Freund were supported by the Air Force Office of Scientific Research [Grant FA9550-19-1-0240].
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