arXiv subjects

Thomas Schmelzer

Publications and source records attributed to Thomas Schmelzer.

9 recordsLinked to original sources

arXiv2026

Goldfarb-Idnani Revisited:Invariants, Certificates, and the Limits of Guessing

The dual active-set method of Goldfarb and Idnani solves the strictly convex quadratic program by adding and dropping one constraint at a time, requiring no phase one. Primal--dual active set and block principal pivoting skip the walk altogether: they guess an entire active set at once and repair it from returning signs. For bound constraints the guess is safe. The system solved on a candidate set is a principal submatrix of $G^{-1}$, positive definite regardless of the guess; this $P$-matrix property guarantees finite termination. For $C^\top x \ge b$ that hypothesis fails, which is our central focus. The working-set system is $C_{\mathcal{A}}^\top G^{-1} C_{\mathcal{A}}$, positive definite only when $C_{\mathcal{A}}$ has full column rank, a property of the guess, not the data. The $P$-matrix property is lost and the structural obstruction invalidates the guarantee. Strict convexity keeps the method usable: KKT conditions make candidate sets certified rather than trusted. Over 600 random instances across four constraint families this failure never occurs. However, duplicating columns of $C$ raises failure rates to 93\%, dropping certified fraction from 100\% to zero. On real long-only portfolio data, the active set reaches 442 of $494 + 1$ constraints.

math.OC↗

arXiv2026

Non-Negative Conjugate Gradients

The conjugate gradient method (CG) solves a symmetric positive definite (SPD) system $Ax=b$, but does not respect $x\geq0$. We develop non-negative conjugate gradients, a solver for the bound-constrained quadratic $\min_x \tfrac12 x^\top A x - b^\top x$ subject to $x\geq0$, $Bx=c$, $A\succ0$, with $Bx=c$ present only in the equality-augmented variant. The solver wraps CG in a primal-dual active-set loop: each step solves the unconstrained system over the free variables, drops variables that turn negative, re-admits any bound variable with negative reduced gradient, and restarts CG. These toggles are the principal pivots of the linear complementarity problem $(A,-b)$, well defined since $A$ is a $P$-matrix. Termination uses the block-principal-pivoting construction of J\'udice and Pires, with a least-index Bland fallback, reaching the unique global minimiser in finitely many outer steps. Our contribution carries this guarantee to the inexact, matrix-free regime the Krylov inner solve requires: once CG's residual is tied to the decision margin, the approximate loop makes the same primal and dual sign decisions as the exact one, and a direct factorisation may replace CG unchanged. Each inner solve is matrix-free: for $A=M^\top M$, the operator $v\mapsto M^\top(Mv)$ needs two products with $M$, no $O(n^2)$ storage, and retains CG's $O(\sqrt\kappa)$ rate, $\kappa=\kappa(A)$. Regularisation or preconditioning that compresses the spectrum lowers the iteration count; a ridge/Tikhonov split $A\mapsto(1-\alpha)A+\alpha R^\top R$ also secures the needed $P$-matrix property. A loop-free projected-gradient reference method, analysed alongside, converges at the slower $O(\kappa)$ rate. On synthetic SPD problems the loop reaches the planted optimum in at most 6 outer steps, and with a direct free-set solve outpaces Lawson--Hanson and an interior-point solver by an order of magnitude on dense instances.

math.OC↗

arXiv2024

A Markowitz Approach to Managing a Dynamic Basket of Moving-Band Statistical Arbitrages

We consider the problem of managing a portfolio of moving-band statistical arbitrages (MBSAs), inspired by the Markowitz optimization framework. We show how to manage a dynamic basket of MBSAs, and illustrate the method on recent historical data, showing that it can perform very well in terms of risk-adjusted return, essentially uncorrelated with the market.

econ.EM↗

arXiv2024

Finding Moving-Band Statistical Arbitrages via Convex-Concave Optimization

We propose a new method for finding statistical arbitrages that can contain more assets than just the traditional pair. We formulate the problem as seeking a portfolio with the highest volatility, subject to its price remaining in a band and a leverage limit. This optimization problem is not convex, but can be approximately solved using the convex-concave procedure, a specific sequential convex programming method. We show how the method generalizes to finding moving-band statistical arbitrages, where the price band midpoint varies over time.

econ.EM↗

arXiv2024

Markowitz Portfolio Construction at Seventy

More than seventy years ago Harry Markowitz formulated portfolio construction as an optimization problem that trades off expected return and risk, defined as the standard deviation of the portfolio returns. Since then the method has been extended to include many practical constraints and objective terms, such as transaction cost or leverage limits. Despite several criticisms of Markowitz's method, for example its sensitivity to poor forecasts of the return statistics, it has become the dominant quantitative method for portfolio construction in practice. In this article we describe an extension of Markowitz's method that addresses many practical effects and gracefully handles the uncertainty inherent in return statistics forecasting. Like Markowitz's original formulation, the extension is also a convex optimization problem, which can be solved with high reliability and speed.

q-fin.PM↗

arXiv2023

A Simple Method for Predicting Covariance Matrices of Financial Returns

We consider the well-studied problem of predicting the time-varying covariance matrix of a vector of financial returns. Popular methods range from simple predictors like rolling window or exponentially weighted moving average (EWMA) to more sophisticated predictors such as generalized autoregressive conditional heteroscedastic (GARCH) type methods. Building on a specific covariance estimator suggested by Engle in 2002, we propose a relatively simple extension that requires little or no tuning or fitting, is interpretable, and produces results at least as good as MGARCH, a popular extension of GARCH that handles multiple assets. To evaluate predictors we introduce a novel approach, evaluating the regret of the log-likelihood over a time period such as a quarter. This metric allows us to see not only how well a covariance predictor does over all, but also how quickly it reacts to changes in market conditions. Our simple predictor outperforms MGARCH in terms of regret. We also test covariance predictors on downstream applications such as portfolio optimization methods that depend on the covariance matrix. For these applications our simple covariance predictor and MGARCH perform similarly.

econ.EM↗

arXiv2020

Constructing trading strategy ensembles by classifying market states

Rather than directly predicting future prices or returns, we follow a more recent trend in asset management and classify the state of a market based on labels. We use numerous standard labels and even construct our own ones. The labels rely on future data to be calculated, and can be used a target for training a market state classifier using an appropriate set of market features, e.g. moving averages. The construction of those features relies on their label separation power. Only a set of reasonable distinct features can approximate the labels. For each label we use a specific neural network to classify the state using the market features from our feature space. Each classifier gives a probability to buy or to sell and combining all their recommendations (here only done in a linear way) results in what we call a trading strategy. There are many such strategies and some of them are somewhat dubious and misleading. We construct our own metric based on past returns but penalising for a low number of transactions or small capital involvement. Only top score-performance-wise trading strategies end up in final ensembles. Using the Bitcoin market we show that the strategy ensembles outperform both in returns and risk-adjusted returns in the out-of-sample period. Even more so we demonstrate that there is a clear correlation between the success achieved in the past (if measured in our custom metric) and the future.

q-fin.TR↗

arXiv2013

Seven Sins in Portfolio Optimization

Although modern portfolio theory has been in existence for over 60 years, fund managers often struggle to get its models to produce reliable portfolio allocations without strongly constraining the decision vector by tight bands of strategic allocation targets. The two main root causes to this problem are inadequate parameter estimation and numerical artifacts. When both obstacles are overcome, portfolio models yield excellent allocations. In this paper, which is primarily aimed at practitioners, we discuss the most common mistakes in setting up portfolio models and in solving them algorithmically.

q-fin.PM↗

arXiv2013

Regression techniques for Portfolio Optimisation using MOSEK

Regression is widely used by practioners across many disciplines. We reformulate the underlying optimisation problem as a second-order conic program providing the flexibility often needed in applications. Using examples from portfolio management and quantitative trading we solve regression problems with and without constraints. Several Python code fragments are given. The code and data are available online at http://www.github.com/tschm/MosekRegression.

q-fin.PM↗