arXiv subjects

Luis F. Zuluaga

Publications and source records attributed to Luis F. Zuluaga.

At least 19 recordsLinked to original sources

arXiv2026

Characterizing QUBO Reformulations of the Max-k-Cut Problem for Quantum Computing

Quantum computing offers significant potential for solving NP-hard combinatorial (optimization) problems that are beyond the reach of classical computers. One way to tap into this potential is by reformulating combinatorial problems as a quadratic unconstrained binary optimization (QUBO) problem. The solution of the QUBO reformulation can then be addressed using adiabatic quantum computing devices or appropriate quantum computing algorithms on gate-based quantum computing devices. In general, QUBO reformulations of combinatorial problems can be readily obtained by properly penalizing the violation of the problem's constraints in the original problem's objective. However, characterizing tight (i.e., minimal but sufficient) penalty coefficients for this purpose is important and non-trivial for enabling the solution of the resulting QUBO in current and near-term quantum computing devices. Along these lines, we present closed-form characterizations of tight penalty coefficients for two distinct QUBO reformulations of the max $k$-cut problem whose values depend on the (weighted) degree of the vertices of the graph defining the problem. These findings contribute to the ongoing effort to make quantum computing a viable tool for solving combinatorial problems at scale. We support our theoretical results with illustrative examples and simple numerical results.

quant-ph↗

arXiv2026

FTCircuitBench: A Benchmark Suite for Fault-Tolerant Quantum Compilation and Architecture

Realizing large-scale quantum advantage is expected to require quantum error correction (QEC), making the compilation and optimization of logical operations a critical area of research. Logical computation imposes distinct constraints and operational paradigms that differ from those of the Noisy Intermediate-Scale Quantum (NISQ) regime, motivating the continued evolution of compilation tools. Given the complexity of this emerging stack, where factors such as gate decomposition precision and computational models must be co-designed, standardized benchmarks and toolkits are valuable for evaluating progress. To support this need, we introduce FTCircuitBench, which serves as: (1) a benchmark suite of impactful quantum algorithms, featuring pre-compiled instances in both Clifford+T and Pauli Based Computation models; (2) a modular end-to-end pipeline allowing users to compile and decompose algorithms for various fault-tolerant architectures, supporting both prebuilt and custom optimization passes; and (3) a toolkit for evaluating the impact of algorithms and optimization across the full compilation stack, providing detailed numerical analysis at each stage. FTCircuitBench is fully open-sourced and maintained on Github.

quant-ph↗

arXiv2026

Computational Framework for Bézier Distributions

Flexible continuous univariate distributions with bounded support are essential for accurate input modeling in stochastic simulation and decision analysis. Although Bézier distributions provide a powerful family capable of representing complex shapes, their adoption has been hindered by the lack of efficient fitting procedures and modern software implementations. This paper develops a computational framework for fitting Bézier distributions to empirical data via both minimum error and maximum likelihood estimation, leveraging first-order optimization methods and exploiting the geometry of the parameter space. We identify provably (asymptotically) lossless convex restrictions of the feasible set that enable efficient projection operators based on isotonic regression and develop first-order algorithms that reduce computational runtime by three to four orders of magnitude compared to traditional derivative-free methods, while delivering consistent fits across real-world data. When benchmarked against the nonlinear solver IPOPT, our methods prove three orders of magnitude faster on average and more robust, while achieving comparable accuracy. To bridge the gap between theory and practice, we introduce bezierv, an open-source Python package providing a unified interface for fitting, analyzing, and convolving Bézier distributions.

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

A new characterization of symmetric $H^+$-tensors and $M$-tensors

In this work, we present a new characterization of symmetric $H^+$-tensors. It is known that a symmetric tensor is an $H^+$-tensor if and only if it is a generalized diagonally dominant tensor with nonnegative diagonal elements. By exploring the diagonal dominance property, we derive new necessary and sufficient conditions for a symmetric tensor to be an $H^+$-tensor. Based on these conditions, we propose a novel method that allows to check if a tensor is a symmetric $H^+$-tensor in polynomial time. Moreover, these results can be applied to the closely related and important class of $M$-tensors. In particular, this allows to efficiently compute the minimum $H$-eigenvalue of symmetric $M$-tensors. Furthermore, we show how this latter result can be used to provide tighter lower bounds for the minimum $H$-eigenvalue of the Fan product of two symmetric $M$-tensors.

math.SP↗

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↗

arXiv2023

Quantum Interior Point Methods for Semidefinite Optimization

We present two quantum interior point methods for semidefinite optimization problems, building on recent advances in quantum linear system algorithms. The first scheme, more similar to a classical solution algorithm, computes an inexact search direction and is not guaranteed to explore only feasible points; the second scheme uses a nullspace representation of the Newton linear system to ensure feasibility even with inexact search directions. The second is a novel scheme that might seem impractical in the classical world, but it is well-suited for a hybrid quantum-classical setting. We show that both schemes converge to an optimal solution of the semidefinite optimization problem under standard assumptions. By comparing the theoretical performance of classical and quantum interior point methods with respect to various input parameters, we show that our second scheme obtains a speedup over classical algorithms in terms of the dimension of the problem $n$, but has worse dependence on other numerical parameters.

quant-ph↗

arXiv2023

On relaxations of the max $k$-cut problem formulations

A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max $k$-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max $k$-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max $k$-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.

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↗

arXiv2021

Revenue Adequate Prices for Chance-Constrained Electricity Markets with Variable Renewable Energy Sources

In a commodity market, revenue adequate prices refer to compensations that ensure that a market participant has a non-negative profit. In this article, we study the problem of deriving revenue adequate prices for an electricity market-clearing model with uncertainties resulting from the use of variable renewable energy sources (VRES). To handle the uncertain nature of the problem, we use a chance-constrained optimization (CCO) approach, which has recently become very popular choice when constructing dispatch electricity models with penetration of VRES (or other sources of uncertainty). Then, we show how prices that satisfy revenue adequacy in expectation for the market administrator, and cost recovery in expectation for all conventional and VRES generators, can be obtained from the optimal dual variables associated with the deterministic equivalent of the CCO market-clearing model. These results constitute a novel contribution to the research of research on revenue adequate, equilibrium, and other types of pricing schemes that have been derived in the literature when the market uncertainties are modeled using stochastic or robust optimization approaches. Unlike in the stochastic approach, the CCO market-clearing model studied here produces uncertainty uniform real-time prices that do not depend on the real-time realization of the VRES generation outcomes. To illustrate our results, we consider a case study electricity market, and contrast the market prices obtained using a revenue adequate stochastic approach and the proposed revenue adequate CCO approach.

econ.TH↗

arXiv2021

Characterization of QUBO reformulations for the maximum $k$-colorable subgraph problem

Quantum devices can be used to solve constrained combinatorial optimization (COPT) problems thanks to the use of penalization methods to embed the COPT problem's constraints in its objective to obtain a quadratic unconstrained binary optimization (QUBO) reformulation of the COPT. However, the particular way in which this penalization is carried out, affects the value of the penalty parameters, as well as the number of additional binary variables that are needed to obtain the desired QUBO reformulation. In turn, these factors substantially affect the ability of quantum computers to efficiently solve these constrained COPT problems. This efficiency is key towards the goal of using quantum computers to solve constrained COPT problems more efficiently than with classical computers. Along these lines, we consider an important constrained COPT problem; namely, the maximum $k$-colorable subgraph (M$k$CS) problem, in which the aim is to find an induced $k$-colorable subgraph with maximum cardinality in a given graph. This problem arises in channel assignment in spectrum sharing networks, VLSI design, human genetic research, and cybersecurity. We derive two QUBO reformulations for the M$k$CS problem, and fully characterize the range of the penalty parameters that can be used in the QUBO reformulations. Further, one of the QUBO reformulations of the M$k$CS problem is obtained without the need to introduce additional binary variables. To illustrate the benefits of obtaining and characterizing these QUBO reformulations, we benchmark different QUBO reformulations of the M$k$CS problem by performing numerical tests on D-Wave's quantum annealing devices. These tests also illustrate the numerical power gained by using the latest D-Wave's quantum annealing device.

quant-ph↗

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↗

arXiv2018

Globally solving Non-Convex Quadratic Programs via Linear Integer Programming techniques

Quadratic programming (QP) is a well-studied fundamental NP-hard optimization problem which optimizes a quadratic objective over a set of linear constraints. In this paper, we reformulate QPs as a mixed-integer linear problem (MILP). This is done via the reformulation of QP as a linear complementary problem, and the use of binary variables and big-M constraints, to model the complementary constraints. To obtain such reformulation, we show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems. Reformulating non-convex QPs as MILPs provides an advantageous way to obtain global solutions as it allows the use of current state-of-the-art MILP solvers. To illustrate this, we compare the performance of our solution approach, labeled quadprogIP, with the current benchmark global QP solver quadprogBB, as well as with BARON, one of the leading non-linear programming (NLP) solvers, and CPLEX's non-convex QP solver, on a large variety of QP test instances. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. For general QPs, quadprogIP outperforms quadprogBB, outperforms BARON in most instances, while CPLEX performs the best on these instances. For box-constrained QPs, quadprogIP has a comparable performance to quadprogBB and BARON in small- to medium-scale instances, but is outperformed by these solvers on large-scale instances; while CPLEX performs the best on box-constrained QP instances. Also, unlike quadprogBB, the solution approach proposed here is able to solve QP instances whose dual feasible set is unbounded. The MATLAB code, called quadprogIP, and the instances used to perform these numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP.

math.OC↗