A new perspective on the complexity of interior point methods for linear programming

In a dynamical systems paradigm, many optimization algorithms are equivalent to applying forward Euler method to the system of ordinary differential equations defined by the vector field of the search directions. Thus the stiffness of such vector fields will play an essential role in the complexity of these methods. We first exemplify this point with a theoretical result for general linesearch methods for unconstrained optimization, which we further employ to investigating the complexity of a primal short-step path-following interior point method for linear programming. Our analysis involves showing that the Newton vector field associated to the primal logarithmic barrier is nonstiff in a sufficiently small and shrinking neighbourhood of its minimizer. Thus, by confining the iterates to these neighbourhoods of the primal central path, our algorithm has a nonstiff vector field of search directions, and we can give a worst-case bound on its iteration complexity. Furthermore, due to the generality of our vector field setting, we can perform a similar (global) iteration complexity analysis when the Newton direction of the interior point method is computed only approximately, using some direct method for solving linear systems of equations

Local quadratic convergence of polynomial-time interior-point methods for conic optimization problems

In this paper, we establish a local quadratic convergence of polynomial-time interior-point methods for general conic optimization problems. The main structural property used in our analysis is the logarithmic homogeneity of self-concordant barrier functions. We propose new path-following predictor-corrector schemes which work only in the dual space. They are based on an easily computable gradient proximity measure, which ensures an automatic transformation of the global linear rate of convergence to the local quadratic one under some mild assumptions. Our step-size procedure for the predictor step is related to the maximum step size (the one that takes us to the boundary). It appears that in order to obtain local superlinear convergence, we need to tighten the neighborhood of the central path proportionally to the current duality gapconic optimization problem, worst-case complexity analysis, self-concordant barriers, polynomial-time methods, predictor-corrector methods, local quadratic convergence

A regularized Interior Point Method for sparse Optimal Transport on Graphs

In this work, the authors address the Optimal Transport (OT) problem on graphs using a proximal stabilized Interior Point Method (IPM). In particular, strongly leveraging on the induced primal-dual regularization, the authors propose to solve large scale OT problems on sparse graphs using a bespoke IPM algorithm able to suitably exploit primal-dual regularization in order to enforce scalability. Indeed, the authors prove that the introduction of the regularization allows to use sparsified versions of the normal Newton equations to inexpensively generate IPM search directions. A detailed theoretical analysis is carried out showing the polynomial convergence of the inner algorithm in the proposed computational framework. Moreover, the presented numerical results showcase the efficiency and robustness of the proposed approach when compared to network simplex solvers

Curvature as a Complexity Bound in Interior-Point Methods

In this thesis, we investigate the curvature of interior paths as a component of complexity bounds for interior-point methods (IPMs) in Linear Optimization (LO). LO is an optimization paradigm, where both the objective and the constraints of the model are represented by linear relationships of the decision variables. Among the class ofalgorithms for LO, our focus is on IPMs which have been an extremely active research area in the last three decades. IPMs in optimization are unique in the sense that they enjoy the best iteration-complexity bounds which are polynomial in the size of the LO problem. The main objects of our interest in this thesis are two distinct curvature measures in the literature, the geometric and the Sonnevend curvature of the central path. The central path is a fundamental tool for the design and the study of IPMs and we will see both that the geometric and Sonnevend\u27s curvature of the central path are proven to be useful in approaching the iteration-complexity questions in IPMs. While the Sonnevend curvature of the central path has been rigorously shown to determine the iteration-complexity of certain IPMs, the role of the geometric curvature in the literature to explain the iteration-complexity is still not well-understood. The novel approach in this thesis is to explore whether or not there is a relationship between these two curvature concepts aiming to bring the geometric curvature into the picture. The structure of the thesis is as follows. In the first three chapters, we present the basic knowledge of path-following IPMs algorithms and review the literature on Sonnevend\u27s curvature and the geometric curvature of the central path. In Chapter 4, we analyze a certain class ofLO problems and show that the geometric and Sonnevend\u27s curvature for these problems display analogous behavior. In particular, the main result of this chapter states that in order to establish an upper bound for the total Sonnevend curvature of the central path, it is sufficient to consider only the case when the number of inequalities is twice as big as the dimension. In Chapter 5, we study the redundant Klee-Minty (KM) construction and prove that the classical polynomial upper bound for IPMs is essentially tight for the Mizuno-Todd-Ye predictor-corrector algorithm. This chapter also provides a negative answer to an open problem about the Sonnevend curvature posed by Stoer et al. in 1993. Chapter 6 investigates a condition number relevant to the Sonnevend curvature and yields a strongly polynomial bound for that curvature in some special cases. Chapter 7 deals with another self-concordant barrier function, the volumetric barrier, and the volumetric path. That chapter investigates some of the basic properties of the volumetric path and shows that certain fundamental properties of the central path failto hold for the volumetric path. Chapter 8 concludes the thesis by providing some final remarks and pointing out future research directions

End-to-end resource analysis for quantum interior point methods and portfolio optimization

We study quantum interior point methods (QIPMs) for second-order cone programming (SOCP), guided by the example use case of portfolio optimization (PO). We provide a complete quantum circuit-level description of the algorithm from problem input to problem output, making several improvements to the implementation of the QIPM. We report the number of logical qubits and the quantity/depth of non-Clifford T-gates needed to run the algorithm, including constant factors. The resource counts we find depend on instance-specific parameters, such as the condition number of certain linear systems within the problem. To determine the size of these parameters, we perform numerical simulations of small PO instances, which lead to concrete resource estimates for the PO use case. Our numerical results do not probe large enough instance sizes to make conclusive statements about the asymptotic scaling of the algorithm. However, already at small instance sizes, our analysis suggests that, due primarily to large constant pre-factors, poorly conditioned linear systems, and a fundamental reliance on costly quantum state tomography, fundamental improvements to the QIPM are required for it to lead to practical quantum advantage.Comment: 38 pages, 15 figure