Homotopy Algorithms Research Articles

Online graph topology learning from matrix-valued time series

The focus is on the statistical analysis of matrix-valued time series, where data is collected over a network of sensors, typically at spatial locations, over time. Each sensor records a vector of features at each time point, creating a vectorial time series for each sensor. The goal is to identify the dependency structure among these sensors and represent it with a graph. When only one feature per sensor is observed, vector auto-regressive (VAR) models are commonly used to infer Granger causality, resulting in a causal graph. The first contribution extends VAR models to matrix-variate models for the purpose of graph learning. Additionally, two online procedures are proposed for both low and high dimensions, enabling rapid updates of coefficient estimates as new samples arrive. In the high-dimensional setting, a novel Lasso-type approach is introduced, and homotopy algorithms are developed for online learning. An adaptive tuning procedure for the regularization parameter is also provided. Given that the application of auto-regressive models to data typically requires detrending, which is not feasible in an online context, the proposed AR models are augmented by incorporating trend as an additional parameter, with a particular focus on periodic trends. The online algorithms are adapted to these augmented data models, allowing for simultaneous learning of the graph and trend from streaming samples. Numerical experiments using both synthetic and real data demonstrate the effectiveness of the proposed methods.

Solving determinantal systems using homotopy techniques

Let K be a field of characteristic zero and let K‾ be an algebraic closure of K. Consider a sequence of polynomials G=(g1,…,gs) in K[X1,…,Xn] with s<n, a polynomial matrix F=[fi,j]∈K[X1,…,Xn]p×q, with p≤q and n=q−p+s+1, and the algebraic set Vp(F,G) of points in K‾ at which all polynomials in G and all p-minors of F vanish. Such polynomial systems appear naturally in polynomial optimization or computational geometry.We provide bounds on the number of isolated points in Vp(F,G) depending on the maxima of the degrees in rows (resp. columns) of F and we design probabilistic homotopy algorithms for computing those points. These algorithms take advantage of the determinantal structure of the system defining Vp(F,G). In particular, the algorithms run in time that is polynomial in the bound on the number of isolated points.

Binary MIMO Detection via Homotopy Optimization and Its Deep Adaptation

In this paper we consider maximum-likelihood (ML) MIMO detection under one-bit quantized observations and binary symbol constellations. This problem is motivated by the recent interest in adopting coarse quantization in massive MIMO systems--as an effective way to scale down the hardware complexity and energy consumption. Classical MIMO detection techniques consider unquantized observations, and many of them are not applicable to the one-bit MIMO case. We develop a new non-convex optimization algorithm for the one-bit ML MIMO detection problem, using a strategy called homotopy optimization. The idea is to transform the ML problem into a sequence of approximate problems, from easy (convex) to hard (close to ML), and with each problem being a gradual modification of its previous. Then, our attempt is to iteratively trace the solution path of these approximate problems. This homotopy algorithm is well suited to the application of deep unfolding, a recently popular approach for turning certain model-based algorithms into data-driven, and performance enhanced, ones. While our initial focus is on one-bit MIMO detection, the proposed technique also applies naturally to the classical unquantized MIMO detection. We performed extensive simulations and show that the proposed homotopy algorithms, both non-deep and deep, have satisfactory bit-error probability performance compared to many state-of-the-art algorithms. Also, the deep homotopy algorithm has attractively low computational complexity.

Homotopy Approach for Energy-Efficient MIMO Systems With Zero-Forcing Transceivers

This paper addresses energy efficiency (EE) optimization problems for uplink multiuser multiple-input–multiple-output (MIMO) systems with zero-forcing (ZF) transceivers, subject to a joint power and rate constraint for each user. As the optimization problem considered in this paper is quite different from the traditional one, directly applying the Dinkelbach method to the new EE optimization problem cannot bring an advantage of low complexity. Hence, we try to propose a novel approach inspired by the homotopy algorithms, where we concentrate our attention on the solution path for the EE optimization. Making use of the piecewise-defined property between power allocations and the overall transmission power, we fit the entire path of solution for every value of the overall transmission power by replacing the search variable with a specially selected parameter. By checking only a finite number of breakpoints, we then try to propose a new homotopy-type algorithm to obtain the maximum EE in an exact closed form to reduce the computational complexity significantly. Simulation results verify the performance of the new method.

Computing Mixed Volume and All Mixed Cells in Quermassintegral Time

The mixed volume counts the roots of generic sparse polynomial systems. Mixed cells are used to provide starting systems for homotopy algorithms that can find all those roots, and track no unnecessary path. Up to now, algorithms for that task were of enumerative type, with no general non- exponential complexity bound. A geometric algorithm is introduced in this paper. Its complexity is bounded in the average and probability-one settings in terms of some geometric invariants: quermassintegrals associated to the tuple of convex hulls of the support of each polynomial. Besides the complexity bounds, numerical results are reported. Those are consistent with an output- sensitive running time for each benchmark family where data is available. For some of those families, an asymptotic running time gain over the best code available at this time was noticed.

Computing Tensor Eigenvalues via Homotopy Methods

We introduce the concept of mode-k generalized eigenvalues and eigenvectors of a tensor and prove some properties of such eigenpairs. In particular, we derive an upper bound for the number of equivalence classes of generalized tensor eigenpairs using mixed volume. Based on this bound and the structures of tensor eigenvalue problems, we propose two homotopy continuation type algorithms to solve tensor eigenproblems. With proper implementation, these methods can find all equivalence classes of isolated generalized eigenpairs and some generalized eigenpairs contained in the positive dimensional components (if there are any). We also introduce an algorithm that combines a heuristic approach and a Newton homotopy method to extract real generalized eigenpairs from the found complex generalized eigenpairs. A MATLAB software package TenEig has been developed to implement these methods. Numerical results are presented to illustrate the effectiveness and efficiency of TenEig for computing complex or real generalized eigenpairs.

Adaptive reweighting homotopy algorithms applied to beamforming

We develop adaptive beamforming algorithms that are robust against sensor failure and ill-conditioning in the autocorrelation matrix (common in low-rank interference scenarios). Both goals are achieved simultaneously through the use of l1 regularization. The algorithms are based on the complex adaptive reweighting homotopy technique. We also develop iterative versions of the algorithms that take advantage of properties of homotopy l1 solvers and dichotomous coordinate iterations to reduce considerably the computational complexity, compared with other regularization methods.

An Alternative and User Friendly Algorithm for the Calibration of Complex DSGE Models

The aim of the following paper is to present a derivative based algorithm, which is able to solve calibration problems for complicated DSGE models. Our algorithm is based on considerations that were mainly done during the 80s, when the computable applied general equilibrium (CAGE) literature began to develop models that could no longer be solved analytically. To solve the resulting problem, researchers began to explore the field of homotopy algorithms. With the rise of the dynamic stochastic general equilibrium (DSGE) models, however, the focus began to shift away from these methods, since the determination of the steady state values for the early DSGE models could be done by hand. Nowadays, as the DSGE models get richer and solutions techniques more advanced, these methods come again to the fore. The usage of complicated frictions in a model that additionally features a fully-fledged banking sector forms on the one hand a brilliant framework for modern policy advice. But on the other hand, it is a complicated mathematical construction, which can in many cases no longer be solved analytically. Since these models start to get more influential, new algorithms need to be developed that can support the researcher in the calibration of those models. Such an algorithm is developed in this paper.

A Globally Convergent Probability-One Homotopy for Linear Programs with Linear Complementarity Constraints

A solution of the standard formulation of a linear program with linear complementarity constraints (LPCC) does not satisfy a constraint qualification. A family of relaxations of an LPCC, associated with a probability-one homotopy map, proposed here is shown to have several desirable properties. The homotopy map is nonlinear, replacing all the constraints with nonlinear relaxations of NCP functions. Under mild existence and rank assumptions, (1) the LPCC relaxations RLPCC($\lambda$) have a solution for $0\le\lambda \le1$; (2) RLPCC(1) is equivalent to LPCC; (3) the Kuhn--Tucker constraint qualification is satisfied at every local or global solution of RLPCC($\lambda$) for almost all $ 0\le\lambda <1$; (4) a point is a local solution of RLPCC(1) (and LPCC) if and only if it is a Kuhn--Tucker point for RLPCC(1); and (5) a homotopy algorithm can find a Kuhn--Tucker point for RLPCC(1). Since the homotopy map is a globally convergent probability-one homotopy, robust and efficient numerical algorithms exist to f...

On LARS/Homotopy Equivalence Conditions for Over-Determined LASSO

We revisit the positive cone condition given by Efron <etal xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <citerefgrp xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <citeref refid="ref1"/></citerefgrp> for the over-determined least absolute shrinkage and selection operator (LASSO). It is a sufficient condition ensuring that the number of nonzero entries in the solution vector keeps increasing when the penalty parameter decreases, based on which the least angle regression (LARS) <citerefgrp xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><citeref refid="ref1"/></citerefgrp> and homotopy <citerefgrp xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <citeref refid="ref2"/></citerefgrp> algorithms yield the same iterates. We show that the positive cone condition is equivalent to the diagonal dominance of the Gram matrix inverse, leading to a simpler way to check the positive cone condition in practice. Moreover, we elaborate on a connection between the positive cone condition and the mutual coherence condition given by Donoho and Tsaig <citerefgrp xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><citeref refid="ref3"/></citerefgrp> , ensuring the exact recovery of any <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$k$</tex> </formula> -sparse representation using both LARS and homotopy.

Stability verification for monotone systems using homotopy algorithms

A monotone self-mapping of the nonnegative orthant induces a monotone discrete-time dynamical system which evolves on the same orthant. If with respect to this system the origin is attractive then there must exist points whose image under the monotone map is strictly smaller than the original point, in the component-wise partial ordering. Here it is shown how such points can be found numerically, leading to a recipe to compute order intervals that are contained in the region of attraction and where the monotone map acts essentially as a contraction. An important application is the numerical verification of so-called generalized small-gain conditions that appear in the stability theory of large-scale systems.

Probability-one homotopy maps for mixed complementarity problems

Probability-one homotopy algorithms have strong convergence characteristics under mild assumptions. Such algorithms for mixed complementarity problems (MCPs) have potentially wide impact because MCPs are pervasive in science and engineering. A probability-one homotopy algorithm for MCPs was developed earlier by Billups and Watson based on the default homotopy mapping. This algorithm had guaranteed global convergence under some mild conditions, and was able to solve most of the MCPs from the MCPLIB test library. This paper extends that work by presenting some other homotopy mappings, enabling the solution of all the remaining problems from MCPLIB. The homotopy maps employed are the Newton homotopy and homotopy parameter embeddings.

Homotopy Methods to Compute Equilibria in Game Theory

This paper presents a complete survey of the use of homotopy methods in game theory. Homotopies allow for a robust computation of game-theoretic equilibria and their refinements. Homotopies are also suitable to compute equilibria that are selected by various selection theories. We present all relevant techniques underlying homotopy algorithms. We give detailed expositions of the Lemke-Howson algorithm and the Van den Elzen-Talman algorithm to compute Nash equilibria in 2-person games, and the Herings-Van den Elzen, Herings-Peeters, and McKelvey-Palfrey algorithms to compute Nash equilibria in general n-person games.

Polyhedral function constrained optimization problems

Recently polyhedral functions have proved distinctly useful in expressing selection criteria in various model building techniques. Here they play the role of a constraint on an estimation problem. Whereas they can always be replaced by an appropriate family of linear constraints, the resulting set can be a very large. Compact representations are available and their use is illustrated by developing both active set and homotopy algorithms for the general polyhedral constrained problem. These are illustrated using some well known data sets.

Numeric vs. symbolic homotopy algorithms in polynomial system solving: a case study

We consider a family of polynomial systems which arises in the analysis of the stationary solutions of a standard discretization of certain semi-linear second-order parabolic partial differential equations. We prove that this family is well-conditioned from the numeric point of view, and ill-conditioned from the symbolic point of view. We exhibit a polynomial-time numeric algorithm solving any member of this family, which significantly contrasts the exponential behavior of all known symbolic algorithms solving a generic instance of this family of systems.

Generalized Linear Product Homotopy Algorithms and the Computation of Reachable Surfaces

In this paper, we apply a homotopy algorithm to the problem of finding points in a moving body that lie on specific algebraic surfaces for a given set of spatial configurations of the body. This problem is a generalization of Burmester’s determination of points in a body that lie on a circle for five planar positions. We focus on seven surfaces that we term “reachable” because they correspond to serial chains with two degree-of-freedom positioning structures combined with a three degree-of-freedom spherical wrist. A homotopy algorithm based on generalized linear products is used to provide a convenient estimate of the number of solutions of these polynomial systems. A parallelized version of this algorithm was then used to numerically determine all of the solutions.

Computing Dynamic Output Feedback Laws

The pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed-loop system has a desired set of eigenvalues. Converting this problem into a question of enumerative geometry, efficient numerical homotopy algorithms to solve this problem for general multiple-input-multiple-output systems have been proposed recently. Despite the wider application range of dynamic feedback laws, the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before. In this note, we present symbolic-numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine. We report on numerical experiments with our publicly available software PHCpack and illustrate its application on various control problems from the literature.

Probability-one homotopies in computational science

Probability-one homotopy algorithms are a class of methods for solving nonlinear systems of equations that, under mild assumptions, are globally convergent for a wide range of problems in science and engineering. Convergence theory, robust numerical algorithms, and production quality mathematical software exist for general nonlinear systems of equations, and special cases such as Brouwer fixed point problems, polynomial systems, and nonlinear constrained optimization. Using a sample of challenging scientific problems as motivation, some pertinent homotopy theory and algorithms are presented. The problems considered are analog circuit simulation (for nonlinear systems), reconfigurable space trusses (for polynomial systems), and fuel-optimal orbital rendezvous (for nonlinear constrained optimization). The mathematical software packages HOMPACK90 and POLSYS_PLP are also briefly described.

Computation of the Nash Equilibrium Selected by the Tracing Procedure in N-Person Games

The heart of the equilibrium selection theory of Harsanyi and Selten (1988, A General Theory of Equilibrium Selection in Games, Cambridge, MA: MIT Press) is given by the tracing procedure, a mathematical construction that adjusts arbitrary prior beliefs into equilibrium beliefs. Although the term “procedure” suggests a numerical approach, the tracing procedure itself is a nonconstructive method. In this paper we propose a homotopy algorithm that generates a path of strategies. By using lexicographic pivoting techniques, it can be shown that for the entire class of noncooperative N-person games, the path converges to an approximate Nash equilibrium, even when the starting point or the game is degenerate. The outcome of the algorithm is shown to be arbitrarily close to the equilibrium beliefs proposed by the tracing procedure. Therefore, the algorithm does not compute just any Nash equilibrium, but one with a sound game-theoretic underpinning. Like other homotopy algorithms, it is easily implemented on a computer. Journal of Economic Literature Classification Numbers: C63, C72.

Probability-one homotopy algorithms for solving the coupled Lyapunov equations arising in reduced-order H 2/H ∞ modeling, estimation, and control

Optimal reduced-order modeling, estimation, and control with respect to combined H 2/ H ∞ criteria give rise to coupled Lyapunov and Riccati equations. To develop reliable numerical algorithms for these problems this paper focuses on the coupled Lyapunov equations which appear as a subset of the synthesis equations. In particular, this paper systematically examines the requirements of probability-one homotopy algorithms to guarantee global convergence. Homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family. The main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled Lyapunov equations and variations based upon canonical forms. These results are essential to the probability-one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms.