arXiv subjects

Juan C. Vera

Publications and source records attributed to Juan C. Vera.

At least 19 recordsLinked to original sources

arXiv2026

Squared polynomial approximation kernels for the hypercube: improved error bounds and implications for Lasserre hierarchies

We propose a new family of polynomial approximation kernels for approximating nonnegative polynomials on the hypercube $[-1,1]^n$. Our Kernels produce polynomial sums-of-squares of degree $r$, achieving an $O(\log^3 r/r^2)$ error in the $\ell_1$-norm of the coefficients. This improves on the known error bound $O(1/r)$ from the literature. As a corollary, we obtain an improved convergence rate for the Lasserre hierarchy for polynomial optimization on the hypercube, again improving a known rate by Baldi and Slot from $O(1/r)$ to $O(\log^3 r/r^2)$.

math.OC↗

arXiv2026

Degree Bounds for Positivstellensätze of general semialgebraic sets

Let $p_{\min}$ denote the minimum of a polynomial $p$ over a (general) compact semialgebraic set $S \subseteq \mathbb{R}^n$. A standard way to approximate $p_{\min}$ is via hierarchies built from Positivstellensätze, which certify nonnegativity of polynomials on $S$ using sums of squares or other classes of globally nonnegative polynomials. As the degree of the certificate grows, the values generated by these hierarchies converge asymptotically to $p_{\min}$. A natural question is, then, to determine explicit bounds on the certificate's degree needed to obtain a prescribed $\varepsilon$-approximation to $p_{\min}$, or equivalently certify the positivity of $f:=p - p_{\min} + \varepsilon$ on $S$. We improve the current best degree bounds for Putinar's and Schmüdgen's SOS-Positivstellensatz over $S$. Also, we obtain degree bounds for Krivine--Stengle's and the recently introduced extended-Handelman's $\mathbb{R}_+$-Positivstellensätze over $S$; providing the first explicit degree bounds for linear optimization-based hierarchies over general compact semialgebraic sets. Our approach is based on a lift-and-project construction in which we add new variables to construct an algebraic representation of the distance to the set $S$ using Łojasiewicz's inequality. This lets us lift the problem of certifying the positivity of $f$ on the (complex) set $S$ to the problem of certifying the positivity of a related polynomial $F$ on a higher-dimensional hypercube. By projecting out the added variables, non-negativity certificates for $F$ on the hypercube become non-negativity certificates for $f$ on $S$. Our approach offers a unified methodology to obtain degree bounds for several Positivstellensatz-based hierarchies over general compact sets, narrowing the gap between results for the hypercube (or other simple sets) and more general semialgebraic sets.

math.OC↗

arXiv2026

Lagrangian Reformulation for Nonconvex Optimization: Tailoring Problems to Specialized Solvers

In recent years, there has been a surge of interest in studying different ways to reformulate nonconvex optimization problems, especially those that involve binary variables. This interest surge is due to advancements in computing technologies, such as quantum and Ising devices, as well as improvements in quantum and classical optimization solvers that take advantage of particular formulations of nonconvex problems to tackle their solutions. Our research characterizes the equivalence between equality-constrained nonconvex optimization problems and their Lagrangian relaxation, enabling the aforementioned new technologies to solve these problems. In addition to filling a crucial gap in the literature, our results are readily applicable to many important situations in practice. To obtain these results, we bridge between specific optimization problem characteristics and broader, classical results on Lagrangian duality for general nonconvex problems. Further, our approach takes a comprehensive approach to the question of equivalence between problem formulations. We consider this question not only from the perspective of the problem's objective but also from the viewpoint of its solution. This perspective, often overlooked in existing literature, is particularly relevant for problems featuring continuous and binary variables.

math.OC↗

arXiv2026

Duality of Hoffman constants

We show that a suitable Slater condition implies a duality inequality between the Hoffman constants of the following feasibility problems: $$ \begin{array}{r} Ax-b \in S\\ x \in R \end{array} \qquad\text{ and }\qquad \begin{array}{r} c-A^T y \in R^*\\ y \in S^*. \end{array} $$ where $A\in \mathbb{R}^{m\times n}$, and $R\subseteq \mathbb{R}^n$ and $S\subseteq \mathbb{R}^m$ are reference polyhedral cones, with respective dual cones $R^*\subseteq \mathbb{R}^n$ and $S^*\subseteq \mathbb{R}^m$. Our approach relies on an exact characterization of Hoffman constants and introduces a novel Hoffman duality inequality for polyhedral set-valued mappings. These two fundamental results also yield a striking identity between the Hoffman constants of box-constrained feasibility problems, which feature a similar primal-dual structure with a box and a linear subspace as reference sets. Additionally, we establish a surprising identity between the Hoffman constants of box-constrained feasibility problems and the chi condition measures for weighted least-squares problems

math.OC↗

arXiv2025

Linear Convergence and Error Bounds for Optimization Without Strong Convexity

Many optimization algorithms$\unicode{x2013}$including gradient descent, proximal methods, and operator splitting techniques$\unicode{x2013}$can be formulated as fixed-point iterations (FPI) of continuous operators. When these operators are averaged, convergence to a fixed point is guaranteed when one exists, but the convergence is generally sublinear. Recent results establish linear convergence of FPI for averaged operators under certain conditions. However, such conditions do not apply to common classes of operators, such as those arising in piecewise linear and quadratic optimization problems. In this work, we prove that a local error-bound condition is both necessary and sufficient for the linear convergence of FPI applied to averaged operators. We provide explicit bounds on the convergence rate and show how these relate to the constants in the error-bound condition. Our main result demonstrates that piecewise linear operators satisfy local error bounds, ensuring linear convergence of the associated optimization algorithms. This leads to a general and practical framework for analyzing convergence behavior in algorithms such as ADMM and Douglas-Rachford in the absence of strong convexity. In particular, we obtain convergence rates that are independent of problem data for linear optimization, and depend only on the condition number of the objective for quadratic optimization.

math.OC↗

arXiv2025

Optimization hierarchies for distance-avoiding sets in compact spaces

Witsenhausen's problem asks for the maximum fraction $α_n$ of the $n$-dimensional unit sphere that can be covered by a measurable set containing no pairs of orthogonal points. The best upper bounds for $α_n$ are given by extensions of the Lovász theta number. In this paper, optimization hierarchies based on the Lovász theta number, like the Lasserre hierarchy, are extended to Witsenhausen's problem and similar problems. These hierarchies are shown to converge and are used to compute the best upper bounds for $α_n$ in low dimensions.

math.MG↗

arXiv2025

Low degree sum-of-squares bounds for the stability number: a copositive approach

The stability number of a graph $G$, denoted as $α(G)$, is the maximum size of an independent (stable) set in $G$. Semidefinite programming (SDP) methods, which originated from Lovász's theta number and expanded through lift-and-project hierarchies as well as sums of squares (SOS) relaxations, provide powerful tools for approximating $α(G)$. We build upon the copositive formulation of $α(G)$ and introduce a novel SDP-based hierarchy of inner approximations to the copositive cone COP$_n$, which is derived from structured SOS representations. This hierarchy preserves essential structural properties that are missing in existing approaches, offers an SDP feasibility formulation at each level despite its non-convexity, and converges finitely to $α(G)$. Our results include examples of graph families that require at least $α(G) - 1$ levels for related hierarchies, indicating the tightness of the de Klerk-Pasechnik conjecture. Notably, on those graph families, our hierarchy achieves $α(G)$ in a single step.

math.OC↗

arXiv2025

SDP bounds on the stability number via ADMM and intermediate levels of the Lasserre hierarchy

We consider the Lasserre hierarchy for computing bounds on the stability number of graphs. The semidefinite programs (SDPs) arising from this hierarchy involve large matrix variables and many linear constraints, which makes them difficult to solve using interior-point methods. We propose solving these SDPs using the alternating direction method of multipliers (ADMM). When the second level of the Lasserre hierarchy for a given graph is intractable for the ADMM, we consider an intermediate-level relaxation of the hierarchy. To warm-start the ADMM, we use an optimal solution from the first level of the Lasserre hierarchy, which is equivalent to the well-known Lovász theta function. Additionally, we use this solution to determine which degree two monomials to add in the Lasserre hierarchy relaxation to obtain an intermediate level between 1 and 2. Computational results demonstrate that our approach yields strong bounds on the stability number, which are computable within reasonable running times. We provide the best-known bounds on the stability number of various graphs from the literature.

math.OC↗

arXiv2024

Reducing non-negativity over general semialgebraic sets to non-negativity over simple sets

A non-negativity certificate (NNC) is a way to write a polynomial so that its non-negativity on a semialgebraic set becomes evident. Positivstellensätze (Psätze) guarantee the existence of NNCs. Both, NNCs and Psätze underlie powerful algorithmic techniques for optimization. This paper proposes a universal approach to derive new Psätze for general semialgebraic sets from ones developed for simpler sets, such as a box, a simplex, or the non-negative orthant. We provide several results illustrating the approach. First, by considering Handelman's Positivstellensatz (Psatz) over a box, we construct non-SOS Schmüdgen-type Psätze over any compact semialgebraic set. That is, a family of Psätze that follow the structure of the fundamental Schmüdgen's Psatz, but where instead of SOS polynomials, any class of polynomials containing the non-negative constants can be used, such as SONC, DSOS/SDSOS, hyperbolic or sums of AM/GM polynomials. Secondly, by considering the simplex as the simple set, we derive a sparse Psatz over general compact sets, which does not require any structural assumptions of the set. Finally, by considering Pólya's Psatz over the non-negative orthant, we derive a new non-SOS Psatz over unbounded sets which satisfy some generic conditions. All these results contribute to the literature regarding the use of non-SOS polynomials and sparse NNCs to derive Psätze over compact and unbounded sets. Throughout the article, we illustrate our results with relevant examples and numerical experiments.

math.OC↗

arXiv2021

Linear convergence of the Douglas-Rachford algorithm via a generic error bound condition

We provide new insight into the convergence properties of the Douglas-Rachford algorithm for the problem $\min_x \{f(x)+g(x)\}$, where $f$ and $g$ are convex functions. Our approach relies on and highlights the natural primal-dual symmetry between the above problem and its Fenchel dual $\min_{u} \{ f^*(u) + g_*(u)\}$ where $g_*(u):=g^*(-u)$. Our main development is to show the linear convergence of the algorithm when a natural error bound condition on the Douglas-Rachford operator holds. We leverage our error bound condition approach to show and estimate the algorithm's linear rate of convergence for three special classes of problems. The first one is when $f$ or$g$ and $f^*$ or $g_*$ are strongly convex relative to the primal and dual optimal sets respectively. The second one is when~$f$ and~$g$ are piecewise linear-quadratic functions. The third one is when~$f$ and~$g$ are the indicator functions of closed convex cones. In all three cases the rate of convergence is determined by a suitable measure of well-posedness of the problem. In the conic case, if the two closed convex cones are a linear subspace $L$ and $\mathbb{R}^n_+$, we establish the following stronger {\em finite termination} result: the Douglas-Rachford algorithm identifies the {\em maximum support sets} for $L\cap \mathbb{R}^n_+$ and $L^{\perp}\cap\mathbb{R}^n_+$ in finitely many steps. Our developments have straightforward extensions to the more general linearly constrained problem $\min_{x,y} \{f(x) + g( y):Ax + By = b\}$ thereby highlighting a direct and straightforward relationship between the Douglas-Rachford algorithm and the alternating direction method of multipliers (ADMM).

math.OC↗

arXiv2021

Sparse non-SOS Putinar-type Positivstellensätze

Recently, non-SOS Positivstellensätze for polynomials on compact semialgebraic sets, following the general form of Schmüdgen's Positivstellensatz, have been derived by appropriately replacing the SOS polynomials with other classes of polynomials. An open question in the literature is how to obtain similar results following the general form of Putinar's Positivstellensatz. Extrapolating the algebraic geometry tools used to obtain this type of result in the SOS case fails to answer this question, because algebraic geometry methods strongly use hallmark properties of the class of SOS polynomials, such as closure under multiplication and closure under composition with other polynomials. In this article, using a new approach, we show the existence of Putinar-type Positivstellensätze that are constructed using non-SOS classes of non-negative polynomials, such as SONC, SDSOS and DSOS polynomials. Even not necessarily non-negative classes of polynomials such as sums of arithmetic-mean/geometric-mean polynomials could be used. Furthermore, we show that these certificates can be written with inherent sparsity characteristics. Such characteristics can be further exploited when the sparsity structure of both the polynomial whose non-negativity is being certified and the polynomials defining the semialgebraic set of interest are known. In contrast with related literature focused on exploiting sparsity in SOS Positivstellensätze, these latter results show how to exploit sparsity in a more general setting in which non-SOS polynomials are used to construct the Positivstellensätze.

math.CA↗

arXiv2021

Computing near-optimal Value-at-Risk portfolios using Integer Programming techniques

Value-at-Risk (VaR) is one of the main regulatory tools used for risk management purposes. However, it is difficult to compute optimal VaR portfolios; that is, an optimal risk-reward portfolio allocation using VaR as the risk measure. This is due to VaR being non-convex and of combinatorial nature. In particular, it is well known that the VaR portfolio problem can be formulated as a mixed integer linear program (MILP) that is difficult to solve with current MILP solvers for medium to large-scale instances of the problem. Here, we present an algorithm to compute near-optimal VaR portfolios that takes advantage of this MILP formulation and provides a guarantee of the solution's near-optimality. As a byproduct, we obtain an algorithm to compute tight lower bounds on the VaR portfolio problem that outperform related algorithms proposed in the literature for this purpose. The near-optimality guarantee provided by the proposed algorithm is obtained thanks to the relation between minimum risk portfolios satisfying a reward benchmark and the corresponding maximum reward portfolios satisfying a risk benchmark. These alternate formulations of the portfolio allocation problem have been frequently studied in the case of convex risk measures and concave reward functions. Here, this relationship is considered for general risk measures and reward functions. To illustrate the efficiency of the presented algorithm, numerical results are presented using historical asset returns from the US financial market.

q-fin.PM↗

arXiv2020

Static Hedging of Weather and Price Risks in Electricity Markets

We present the closed-form solution to the problem of hedging price and quantity risks for energy retailers (ER), using financial instruments based on electricity price and weather indexes. Our model considers an ER who is intermediary in a regulated electricity market. ERs buy a fixed quantity of electricity at a variable cost and must serve a variable demand at a fixed cost. Thus ERs are subject to both price and quantity risks. To hedge such risks, an ER could construct a portfolio of financial instruments based on price and weather indexes. We construct the closed form solution for the optimal portfolio for the mean-Var model in the discrete setting. Our model does not make any distributional assumption.

q-fin.MF↗

arXiv2020

Contingent Capital with Stock Price Triggers in Interbank Networks

This paper studies existence and uniqueness of equilibrium prices in a model of the banking sector in which banks trade contingent convertible bonds with stock price triggers among each other. This type of financial product was proposed as an instrument for stabilizing the global banking system after the financial crisis. Yet it was recognized early on that these products may create circularity problems in the definition of stock prices - even in the absence of trade. We find that if conversion thresholds are such that bond holders are indifferent about marginal conversions, there exists a unique equilibrium irrespective of the network structure. When thresholds are lower, existence of equilibrium breaks down while higher thresholds may lead to multiplicity of equilibria. Moreover, there are complex network effects. One bank's conversion may trigger further conversions - or prevent them, depending on the constellations of asset values and conversion triggers.

econ.GN↗

arXiv2020

Equivalence and invariance of the chi and Hoffman constants of a matrix

We show that the following two condition measures of a full column rank matrix $A \in \mathbb{R}^{m\times n}$ are identical: the chi constant and a signed Hoffman constant. This identity is naturally suggested by the evident invariance of the chi constant under sign changes of the rows of $A$. We also show that similar equivalence and invariance properties extend to variants of the chi and Hoffman constants that depend only on the linear subspace $A(\mathbb{R}^n):=\{Ax: x\in\mathbb{R}^n\} \subseteq \mathbb{R}^m$. Finally, we show similar identities between the chi constants and signed versions of Renegar's and Grassmannian condition measures.

math.OC↗

arXiv2019

New bounds for truthful scheduling on two unrelated selfish machines

We consider the minimum makespan problem for $n$ tasks and two unrelated parallel selfish machines. Let $R_n$ be the best approximation ratio of randomized monotone scale-free algorithms. This class contains the most efficient algorithms known for truthful scheduling on two machines. We propose a new $Min-Max$ formulation for $R_n$, as well as upper and lower bounds on $R_n$ based on this formulation. For the lower bound, we exploit pointwise approximations of cumulative distribution functions (CDFs). For the upper bound, we construct randomized algorithms using distributions with piecewise rational CDFs. Our method improves upon the existing bounds on $R_n$ for small $n$. In particular, we obtain almost tight bounds for $n=2$ showing that $|R_2-1.505996|<10^{-6}$.

math.OC↗

arXiv2019

Using Column Generation to Solve Extensions to the Markowitz Model

We introduce a solution scheme for portfolio optimization problems with cardinality constraints. Typical portfolio optimization problems are extensions of the classical Markowitz mean-variance portfolio optimization model. We solve such type of problems using a method similar to column generation. In this scheme, the original problem is restricted to a subset of the assets resulting in a master convex quadratic problem. Then the dual information of the master problem is used in a sub-problem to propose more assets to consider. We also consider other extensions to the Markowitz model to diversify the portfolio selection within the given intervals for active weights.

math.OC↗

arXiv2019

Generalizations of Schoenberg's theorem on positive definite kernels

The seminal theorem of I.J. Schoenberg characterizes positive definite (p.d.) kernels on the unit sphere $S^{n-1}$ invariant under the automorphisms of the sphere. We obtain two generalizations of this theorem for p.d. kernels on fiber bundles. Our first theorem characterizes invariant p.d. kernels on bundles whose fiber is a product of a compact set and the unit sphere. This result implies, in particular, a characterization of invariant under the automorphisms of the sphere, p.d. kernels on a product of $S^{n-1}$ and a compact set. Our second result characterizes invariant p.d. kernels on the bundle whose fiber is $S^{n-1}$, base space is $(S^{n-1})^{r}$ and map is the projection on the base space. This set of kernels is isomorphic to the set of invariant under the automorphisms of the sphere continuous functions $F$ on $(S^{n-1})^{r+2}$ such that $F(\cdot,\cdot ,Z)$ is positive definite for every $Z \in (S^{n-1})^{r}$. When $Z$ is fixed, this class reduces to the class of p.d. kernels invariant under the stabilizer of $Z$ in the automorphism group of the sphere. For $r=1$ these kernels have been used to obtain upper bounds for the spherical codes problem. Our extension for $r>1$ can be used to construct new upper bounds on the size of spherical codes.

math.CA↗