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Roland Hildebrand

Publications and source records attributed to Roland Hildebrand.

At least 19 recordsLinked to original sources
arXiv2026

Norm-Constrained Flows and Sign-Based Optimization: Theory and Algorithms

Sign Gradient Descent (SignGD) uses only the coordinate-wise sign of the gradient. We study this method through norm-constrained continuous-time dynamics: at each point, the velocity is chosen to minimize the directional derivative over a unit norm ball. This recovers the sign flow for the $\ell_\infty$ constraint, normalized gradient flow for $\ell_2$, and greedy coordinate directions for $\ell_1$. We formulate the resulting dynamics as a set-valued differential inclusion, prove existence of solutions, and derive an exact energy identity. This identity also gives finite-time convergence under a Polyak-Lojasiewicz inequality expressed in the dual norm. We then connect the canonical set-valued flow with classical Filippov regularization of discontinuous selectors, which gives a precise description of crossing and sliding near switching sets. Motivated by this behavior, we introduce two face-aware SignGD variants, one-hit freeze and two-hit sliding-track. Both methods modify the usual sign direction by damping selected coordinates when a crossing or persistent switching is detected. We derive descent certificates for these damped sign directions and introduce a safeguard that preserves a global linear convergence rate under $\ell_\infty$-smoothness and a Polyak-Lojasiewicz inequality. For the $\ell_1$ flow, we also introduce convex-combination updates on active faces and prove a corresponding linear convergence guarantee. Finally, we analyze mass-aware face restrictions and an inertial SignGD method with restart. Numerical experiments illustrate the certified step rules, the face-aware variants, and the inertial scheme.

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arXiv2026

Sufficient conditions for the existence of exponential-polynomial expansions for solutions of certain differential equations

We consider ordinary differential equations (ODE) of the form $u''u - (u')^2 = e^{-x}P(u) - 1$, where $P$ is a polynomial. In previous work, necessary conditions on $P$ have been established for certain families of solutions of these ODEs to have asymptotic expansions of the form $u(x) = \sum_{k=0}^{\infty} p_k(x+c)e^{-kx}$ for $Re\,x \to +\infty$, where $c \in \mathbb C$ is an arbitrary constant parameterizing the solution family, and $p_k$ are polynomials, with $p_0(x) = x$. These conditions amount to $P(0) = 0$ and $P'(0) = \frac12P''(0)$. Here we show that these two conditions are also sufficient. The results imply the existence of corresponding expansions for certain degenerate Painlevé III transcendents.

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arXiv2026

Necessary conditions for the existence of exponential-polynomial expansions for solutions of certain differential equations

We consider ordinary differential equations (ODE) of the form $u''u - (u')^2 = e^{-x}P(u) - 1$, where $P$ is a polynomial. For $P = u^k$, $k = 3,4,6$ this ODE is equivalent to certain degenerate Painlevé III equations. We study whether families of solutions of these ODEs have asymptotic expansions of the form $u(x) = \sum_{k=0}^{\infty} p_k(x+c)e^{-kx}$ for $Re\,x \to +\infty$, where $c \in \mathbb C$ is an arbitrary constant parameterizing the solution family, $p_k$ are polynomials, with $p_0(x) = x$. We find necessary conditions on $P$ for such expansions to exist. Numerical experiments suggest that these conditions are also sufficient, and the expansions are not only formal, but actually provide a series representation of the solutions. Numerical evidence also suggests a conjecture on the nonnegativity of coefficients of the $p_k$.

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arXiv2026

IRON: Implicit Resolvent Optimization under Noise

We study stochastic optimization from a joint continuous-discrete point of view. Starting from a second-order stochastic differential equation interpreted as a noisy accelerated gradient flow, we discretize the dynamics by a fully implicit Backward-Euler scheme. This leads to a resolvent, or proximal-type, update, computed in practice through Levenberg-Marquardt, Newton, or trust-region-type inner solves. The resulting method, denoted by $\text{IRON}_{\text{FI}}$, admits a Lyapunov mean-square recursion. The main conclusion is that increasing the implicit stepsize $α$ improves the contraction factor and decreases the stationary mean-square error bound. Under sufficiently accurate inner solves, this bound scales as $O(1/α)$; in particular, for large enough $α$, the recursion is contractive and the stationary error bound vanishes as $α\to\infty$. We establish the theory for smooth strongly convex objectives and provide a sharper quadratic analysis with an explicit stationary constant. The numerical experiments support the theory in the strongly convex case and illustrate the same qualitative behavior in nonconvex and learning settings. To the best of our knowledge, this fully implicit inertial-resolvent discretization, where the noise acts as an additive perturbation of the resolvent center and yields an $O(1/α)$ stationary-MSE law, has not been isolated in this form before. The broader message is that discretization is not only an implementation choice: in stochastic optimization, the numerical integration rule can directly affect the long-time stability of the iterates under noise.

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arXiv2025

Speeding up the Goemans-Williamson randomized procedure by difference-of-convex optimization

We present a novel approach to accelerate the Goemans-Williamson (GW) randomized rounding procedure for quadratic unconstrained binary optimization (QUBO) problems. Instead of solving the conventional semi-definite programming (SDP) relaxation, which is computationally expensive, we employ a difference-of-convex (DC) optimization framework to efficiently approximate the SDP solution. The DC optimization produces candidate vectors that are then used within the GW randomized rounding scheme to generate high-quality binary solutions. Furthermore, we perform direct expectation minimization over manifolds of matrices with limited rank to further enhance the solution quality. Our method is benchmarked on real-world QUBO instances, including inverse kinematics problems, and compared against state-of-the-art solvers, such as quantum-inspired algorithms, demonstrating competitive approximation guarantees alongside substantial computational gains.

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arXiv2025

Numerical analysis of the convex relaxation of the barrier parameter functional of self-concordant barriers

Self-concordant barriers are essential for interior-point algorithms in conic programming. To speed up the convergence it is of interest to find a barrier with the lowest possible parameter for a given cone. The barrier parameter is a non-convex function on the set of self-concordant barriers on a given cone, and finding an optimal barrier amounts to solving a non-convex infinite-dimensional optimization problem. In this work we study the degradation of the optimal value of the problem when the problem is convexified, and provide an estimate of the accuracy of the convex relaxation. The amount of degradation can be computed by comparing a 1-parameter family of non-convex bodies in $R^3$ with their convex hulls. Our study provides insight into the degree of non-convexity of the problem and opens up the possibility of constructing suboptimal barriers by solving the convex relaxation

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arXiv2025

Robustifying networks for flow problems against edge failure

We consider the robust version of a multi-commodity network flow problem. The robustness is defined with respect to the deletion, or failure, of edges. While the flow problem itself is a polynomially-sized linear program, its robust version is a saddle-point problem with discrete variables. We present two approaches for the solution of the robust network flow problem. One way is to formulate the problem as a bigger linear program. The other is to solve a multi-level optimization problem, where the linear programs appearing at the lower level can be solved by the dual simplex method with a warm start. We then consider the problem of robustifying the network. This is accomplished by optimally using a fixed budget for strengthening certain edges, i.e., increasing their capacity. This problem is solved by a sequence of linear programs at the upper level, while at the lower levels the mentioned dual simplex algorithm is employed.

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arXiv2024

Optimization in complex spaces with the Mixed Newton Method

We propose a second-order method for unconditional minimization of functions $f(z)$ of complex arguments. We call it the Mixed Newton Method due to the use of the mixed Wirtinger derivative $\frac{\partial^2f}{\partial\bar z\partial z}$ for computation of the search direction, as opposed to the full Hessian $\frac{\partial^2f}{\partial(z,\bar z)^2}$ in the classical Newton method. The method has been developed for specific applications in wireless network communications, but its global convergence properties are shown to be superior on a more general class of functions $f$, namely sums of squares of absolute values of holomorphic functions. In particular, for such objective functions minima are surrounded by attraction basins, while the iterates are repelled from other types of critical points. We provide formulas for the asymptotic convergence rate and show that in the scalar case the method reduces to the well-known complex Newton method for the search of zeros of holomorphic functions. In this case, it exhibits generically fractal global convergence patterns.

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arXiv2024

Mixed Newton Method for Optimization in Complex Spaces

In this paper, we modify and apply the recently introduced Mixed Newton Method, which is originally designed for minimizing real-valued functions of complex variables, to the minimization of real-valued functions of real variables by extending the functions to complex space. We show that arbitrary regularizations preserve the favorable local convergence properties of the method, and construct a special type of regularization used to prevent convergence to complex minima. We compare several variants of the method applied to training neural networks with real and complex parameters.

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arXiv2024

Half-dimensional immersions into the para-complex projective space and Ruh-Vilms type theorems

In this paper we study isometric immersions $f:M^n \to {\mathbb {C}^{\prime}}\!P^n$ of an $n$-dimensional pseudo-Riemannian manifold $M^n$ into the $n$-dimensional para-complex projective space ${\mathbb {C}^{\prime}}\!P^n$. We study the immersion $f$ by means of a lift $\mathfrak f$ of $f$ into a quadric hypersurface in ${S^{2n+1}_{n+1}}$. We find the frame equations and compatibility conditions. We specialize these results to dimension $n = 2$ and a definite metric on $M^2$ in isothermal coordinates and consider the special cases of Lagrangian surface immersions and minimal surface immersions. We characterize surface immersions with special properties in terms of primitive harmonicity of the Gauss maps.

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arXiv2022

On the structure of the $6 \times 6$ copositive cone

In this work we complement the description of the extreme rays of the $6 \times 6$ copositive cone with some topological structure. In a previous paper we decomposed the set of extreme elements of this cone into a disjoint union of pieces of algebraic varieties of different dimension. In this paper we link this classification to the recently introduced combinatorial characteristic called extended minimal zero support set. We determine those components which are essential, i.e., which are not embedded in the boundary of other components. This allows to drastically decrease the number of cases one has to consider when investigating different properties of the $6 \times 6$ copositive cone. As an application, we construct an example of a copositive $6 \times 6$ matrix with unit diagonal which does not belong to the Parrilo inner sum of squares relaxation ${\cal K}^{(1)}_6$.

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arXiv2021

Convex optimization

This textbook is based on lectures given by the authors at MIPT (Moscow), HSE (Moscow), FEFU (Vladivostok), V.I. Vernadsky KFU (Simferopol), ASU (Republic of Adygea), and the University of Grenoble-Alpes (Grenoble, France). First of all, the authors focused on the program of a two-semester course of lectures on convex optimization, which is given to students of MIPT. The first chapter of this book contains the materials of the first semester ("Fundamentals of convex analysis and optimization"), the second and third chapters contain the materials of the second semester ("Numerical methods of convex optimization"). The textbook has a number of features. First, in contrast to the classic manuals, this book does not provide proofs of all the theorems mentioned. This allowed, on one side, to describe more themes, but on the other side, made the presentation less self-sufficient. The second important point is that part of the material is advanced and is published in the Russian educational literature, apparently for the first time. Third, the accents that are given do not always coincide with the generally accepted accents in the textbooks that are now popular. First of all, we talk about a sufficiently advanced presentation of conic optimization, including robust optimization, as a vivid demonstration of the capabilities of modern convex analysis.

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arXiv2021

Extremal cubics on the circle and the 2-sphere

We study balls of homogeneous cubics on $\mathbb R^n$, $n = 2,3$, which are bounded by unity on the unit sphere. For $n = 2$ we completely describe the facial structure of this norm ball, while for $n = 3$ we classify all extremal points and describe some families of faces.

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arXiv2021

Improving the constant in Nesterov's $\fracπ{2}$-theorem

One of the hard optimization problems that has a semi-definite relaxation with quantitative bound on the approximation error is the maximization of a convex quadratic form on the hypercube. The relaxation not only yields an upper bound on the optimal value, but its solution can be used to construct random sub-optimal solutions of the original problem whose expected value is not less than $\frac{2}π$ times the value of the relaxation. This constant cannot be improved globally. More precisely, for every $ε> 0$ there exists a problem instance for which the ratio of the two values in question is larger than $\fracπ{2} - ε$. However, if a given problem instance is considered, then the relaxation yields a concrete solution which may result in a much better ratio. In this contribution we present an improved, explicit bound depending on the rank of the solution. We consider also the problem of maximization of a convex hermitian quadratic form on the complex poly-disc. In this case a bound on the approximation error is given by the $\frac{4}π$-theorem of Ben-Tal, Nemirovski, and Roos. The derivation of a rank-dependent improved bound is similar to the real case. In the complex case we provide explicit expressions in the form of an infinite series and conjecture a closed-form expression.

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arXiv2021

Semi-definite representations for sets of cubics on the 2-sphere

The compact set of homogeneous quadratic polynomials in $n$ real variables with modulus bounded by 1 on the unit sphere $S^{n-1}$ is trivially semi-definite representable. The compact set of homogeneous ternary quartics with modulus bounded by 1 on the unit sphere $S^2$ is also semi-definite representable. This suggests that the compact set of homogeneous ternary cubics with modulus bounded by 1 on $S^2$ is semi-definite representable. We deduce an explicit semi-definite representation of this norm ball. More generally, we provide a semi-definite description of the cone of inhomogeneous ternary cubics which are nonnegative on $S^2$.

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arXiv2021

Projectively self-concordant barriers

Self-concordance is the most important property required for barriers in convex programming. It is intrinsically linked to the affine structure of the underlying space. Here we introduce an alternative notion of self-concordance which is linked to the projective structure. A function on a set $X \subset A^n$ in an $n$-dimensional affine space is projectively self-concordant if and only if it can be extended to an affinely self-concordant logarithmically homogeneous function on the conic extension $K \subset V^{n+1}$ of $X$ in the $(n+1)$-dimensional vector space obtained by homogenization of $A^n$. The feasible sets in conic programs, notably linear and semi-definite programs, are naturally equipped with projectively self-concordant barriers. However, the interior-point methods used to solve these programs employ only affine self-concordance. We show that estimates used in the analysis of interior-point methods are tighter for projective self-concordance, in particular inner and outer approximations of the set. This opens the way to a better tuning of parameters in interior-points algorithms to allow larger steps and hence faster convergence. Projective self-concordance is also a useful tool in the theoretical analysis of logarithmically homogeneous barriers on cones.

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arXiv2021

Optimal inequalities between distances in convex projective domains

On any proper convex domain in real projective space there exists a natural Riemannian metric, the Blaschke metric. On the other hand, distances between points can be measured in the Hilbert metric. Using techniques of optimal control, we provide inequalities lower bounding the Riemannian length of the line segment joining two points of the domain by the Hilbert distance between these points, thus strengthening a result of Tholozan. Our estimates are valid for a whole class of Riemannian metrics on convex projective domains, namely those induced by convex non-degenerate centro-affine hypersurface immersions. If the immersions are asymptotic to the boundary of the convex cone over the domain, then we can also upper bound the Riemmanian length. On these classes, and in particular for the Blaschke metric, our inequalities are optimal.

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