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We introduce a new quantum optimization algorithm for dense Linear Programming problems, which can be seen as the quantization of the Interior Point Predictor-Corrector algorithm \cite{Predictor-Corrector} using a Quantum Linear System…
Linear programming (LP) is an extremely useful tool and has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such…
In the literature, besides the assumption of strict complementarity, superlinear convergence of implementable polynomial-time interior point algorithms using known search directions, namely, the HKM direction, its dual or the NT direction,…
We propose an inexact infeasible arc-search interior-point method for solving linear optimization problems. The method combines an arc-search strategy with inexact solutions to Newton systems and admits a polynomial iteration complexity…
Interior point methods (IPMs) are a common approach for solving linear programs (LPs) with strong theoretical guarantees and solid empirical performance. The time complexity of these methods is dominated by the cost of solving a linear…
We present a numerical method for the local solution of nonlinear programming problems. The SUMT approach of Fiacco and McCormick results in a merit function with quadratic penalties and logarithmic barriers. Our NLP solver works by…
We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for the largest sparse real and symmetric…
We consider the solution of large-scale nonlinear algebraic Hermitian eigenproblems of the form $T(\lambda)v=0$ that admit a variational characterization of eigenvalues. These problems arise in a variety of applications and are…
In this paper we present a new algorithm for solving linear programs that requires only $\tilde{O}(\sqrt{rank(A)}L)$ iterations to solve a linear program with $m$ constraints, $n$ variables, and constraint matrix $A$, and bit complexity…
Although the classical LQR design method has been very successful in real world engineering designs, in some cases, the classical design method needs modifications because of the saturation in actuators. This modified problem is sometimes…
Quantum Interior Point Methods (QIPMs) have been attracting significant interests recently due to their potential of solving optimization problems substantially faster than state-of-the-art conventional algorithms. In general, QIPMs use…
In this paper we consider the numerical solution to the soft-margin support vector machine optimization problem. This problem is typically solved using the SMO algorithm, given the high computational complexity of traditional optimization…
In this work, we introduce an interior-point method that employs tensor decompositions to efficiently represent and manipulate the variables and constraints of semidefinite programs, targeting problems where the solutions may not be…
We prove that the classic logarithmic barrier problem is equivalent to a particular logarithmic barrier positive relaxation problem with barrier and scaling parameters. Based on the equivalence, a line-search primal-dual interior-point…
The focus in this work is on interior-point methods for inequality-constrained quadratic programs, and particularly on the system of nonlinear equations to be solved for each value of the barrier parameter. Newton iterations give high…
Quantum linear system algorithms (QLSAs) have the potential to speed up algorithms that rely on solving linear systems. Interior Point Methods (IPMs) yield a fundamental family of polynomial-time algorithms for solving optimization…
Since more than three decades, interior-point methods proved very useful for optimization, from linear over semidefinite to conic (and partly beyond non-convex) programming; despite the fact that already in the semidefinite case (even when…
We describe how the low-rank structure in an SDP can be exploited to reduce the per-iteration cost of a convex primal-dual interior-point method down to $O(n^{3})$ time and $O(n^{2})$ memory, even at very high accuracies. A traditional…
Quantum linear system algorithms (QLSA) have the potential to speed up Interior Point Methods (IPM). However, a major challenge is that QLSAs are inexact and sensitive to the condition number of the coefficient matrices of linear systems.…
This article presents a method for solving large-scale linear inverse problems regular- ized with a nonlinear, edge-preserving penalty term such as the total variation or Perona-Malik. In the proposed scheme, the nonlinearity is handled…