arXiv subjects
Publications and source records attributed to Javier Pena.
We develop a few calculus rules to compute or lower bound the facial distance of a polytope. We illustrate our calculus rules on various popular polytopes. In particular, we provide a provide a lower bound on the facial distance of the Birkhoff polytope.
Deterministic global optimization of nonlinear models is important in many scientific and engineering applications. This framework typically involves repeatedly solving convex relaxations of the nonconvex problem, meaning that the strength of the relaxations and the cost of computing them directly determine overall efficiency and solution quality. In this work, we develop a tailored continuous convexification framework for Kolmogorov-Arnold Networks in which the univariate components are polynomial functions. By exploiting the additive separable structure of this architecture, the relaxation problem reduces to computing tight convex envelopes of univariate polynomials. We propose a continuous variant of the classical Graham Scan that constructs these envelopes exactly by identifying the bitangents of the polynomial convex hull without discretization or factorable reformulations. We establish the correctness of the algorithm and characterize its computational complexity, and show how these envelopes can be combined to construct strong convex relaxations for polynomial KANs. Computational results demonstrate that the proposed relaxations are both strong and robust, often producing bounds that are comparable, or even orders of magnitude tighter than relaxations of state-of-the-art global optimization solvers while remaining computationally efficient.
Quantum entanglement lies at the heart of quantum information science, yet its reliable detection in high-dimensional or noisy systems remains a fundamental computational challenge. Semidefinite programming (SDP) hierarchies, such as the Doherty-Parrilo-Spedalieri (DPS) and Extension (EXT) hierarchies, offer complete methods for entanglement detection, but it is well known that their practical use is limited by exponential growth in problem size if implemented naively. We make three contributions. First, we introduce a new SDP hierarchy, PST, that is sandwiched between EXT and DP -- offering a tighter approximation to the set of separable states than EXT, while incurring significantly lower computational overhead than DPS. Second, we explicitly construct compact, polynomially-scalable descriptions of EXT and PST using partition mappings and operators. These descriptions in turn yield formulations that satisfy desirable properties such as the Slater condition and are well-suited to both first-order methods (FOMs) and interior-point methods (IPMs). Third, we design a suite of entanglement detection algorithms: three FOMs (Frank-Wolfe, projected gradient, and fast projected gradient) based on a least-squares formulation, and a custom primal-dual IPM based on a conic programming formulation. These methods are numerically stable and capable of producing entanglement witnesses or proximity measures, even in cases where states lie near the boundary of separability. Numerical experiments on benchmark quantum states demonstrate that our algorithms improve the ability to solve deeper levels of the SDP hierarchy.
We provide a template to derive convergence rates for the following popular versions of the Frank-Wolfe algorithm on polytopes: vanilla Frank-Wolfe, Frank-Wolfe with away steps, Frank-Wolfe with blended pairwise steps, and Frank-Wolfe with in-face directions. Our template shows how the convergence rates follow from two affine-invariant properties of the problem, namely, error bound and extended curvature. These properties depend solely on the polytope and objective function but not on any affine-dependent object like norms. For each one of the above algorithms, we derive rates of convergence ranging from sublinear to linear depending on the degree of the error bound.
Recent papers have shown that the Frank-Wolfe algorithm (FW) with open-loop step-sizes exhibits rates of convergence faster than the iconic $\mathcal{O}(t^{-1})$ rate. In particular, when the minimizer of a strongly convex function over a polytope lies in the relative interior of a feasible region face, the FW with open-loop step-sizes $\eta_t = \frac{\ell}{t+\ell}$ for $\ell \in \mathbb{N}_{\geq 2}$ has accelerated convergence $\mathcal{O}(t^{-2})$ in contrast to the rate $\Omega(t^{-1-\epsilon})$ attainable with more complex line-search or short-step step-sizes. Given the relevance of this scenario in data science problems, research has grown to explore the settings enabling acceleration in open-loop FW. However, despite FW's well-known affine invariance, existing acceleration results for open-loop FW are affine-dependent. This paper remedies this gap in the literature by merging two recent research trajectories: affine invariance (Wirth et al., 2023b) and open-loop step-sizes (Pena, 2021). In particular, we extend all known non-affine-invariant convergence rates for FW with open-loop step-sizes to affine-invariant results.
We show that the conditional gradient method for the convex composite problem \[\min_x\{f(x) + \Psi(x)\}\] generates primal and dual iterates with a duality gap converging to zero provided a suitable {\em growth property} holds and the algorithm makes a judicious choice of stepsizes. The rate of convergence of the duality gap to zero ranges from sublinear to linear depending on the degree of the growth property. The growth property and convergence results depend on the pair $(f,\Psi)$ in an affine invariant and norm-independent fashion.
We propose a simple projection and rescaling algorithm that finds maximum support solutions to the pair of feasibility problems \[ \text{find} \; x\in L\cap\mathbb{R}^n_{+} \;\;\;\; \text{ and } \; \;\;\;\; \text{find} \; \hat x\in L^\perp\cap\mathbb{R}^n_{+}, \] where $L$ is a linear subspace of $\mathbb{R}^n$ and $L^\perp$ is its orthogonal complement. The algorithm complements a basic procedure that involves only projections onto $L$ and $L^\perp$ with a periodic rescaling step. The number of rescaling steps and thus overall computational work performed by the algorithm are bounded above in terms of a condition measure of the above pair of problems. Our algorithm is a natural but significant extension of a previous projection and rescaling algorithm that finds a solution to the problem \[ \text{find} \; x\in L\cap\mathbb{R}^n_{++} \] when this problem is feasible. As a byproduct of our new developments, we obtain a sharper analysis of the projection and rescaling algorithm in the latter special case.
We offer a unified treatment of distinct measures of well-posedness for homogeneous conic systems. To that end, we introduce a distance to infeasibility based entirely on geometric considerations of the elements defining the conic system. Our approach sheds new light on and connects several well-known condition measures for conic systems, including {\em Renegar's} distance to infeasibility, the {\em Grassmannian} condition measure, a measure of the {\em most interior} solution, and other geometric measures of {\em symmetry} and of {\em depth} of the conic system.
We give a characterization of the Hoffman constant of a system of linear constraints in $\R^n$ {\em relative} to a {\em reference polyhedron} $R\subseteq\R^n$. The reference polyhedron $R$ represents constraints that are easy to satisfy such as box constraints. In the special case $R = \R^n$, we obtain a novel characterization of the classical Hoffman constant. More precisely, suppose $R\subseteq \mathbb{R}^n$ is a reference polyhedron, $A\in \R^{m\times n},$ and $A(R):=\{Ax: x\in R\}$. We characterize the sharpest constant $H(A|R)$ such that for all $b \in A(R) + \R^m_+$ and $u\in R$ \[ \dist(u, P_{A}(b)\cap R) \le H(A|R) \cdot \|(Au-b)_+\|, \] where $P_A(b) = \{x\in \R^n:Ax\le b\}$. Our characterization is stated in terms of the largest of a canonical collection of easily computable Hoffman constants. Our characterization in turn suggests new algorithmic procedures to compute Hoffman constants.
This paper documents a computational implementation of a {\em projection and rescaling algorithm} for finding most interior solutions to the pair of feasibility problems \[ \text{find} \; x\in L\cap\mathbb{R}^n_{+} \;\;\;\; \text{ and } \; \;\;\;\; \text{find} \; \hat x\in L^\perp\cap\mathbb{R}^n_{+}, \] where $L$ denotes a linear subspace in $\mathbb{R}^n$ and $L^\perp$ denotes its orthogonal complement. The projection and rescaling algorithm is a recently developed method that combines a {\em basic procedure} involving only low-cost operations with a periodic {\em rescaling step.} We give a full description of a MATLAB implementation of this algorithm and present multiple sets of numerical experiments on synthetic problem instances with varied levels of conditioning. Our computational experiments provide promising evidence of the effectiveness of the projection and rescaling algorithm. Our MATLAB code is publicly available. Furthermore, the simplicity of the algorithm makes a computational implementation in other environments completely straightforward.
We provide new insight into a {\em generalized conditional subgradient} algorithm and a {\em generalized mirror descent} algorithm for the convex minimization problem \[ \min_x \; \{f(Ax) + h(x)\}.\] As Bach showed in [{\em SIAM J. Optim.}, 25 (2015), pp. 115--129], applying either of these two algorithms to this problem is equivalent to applying the other one to its Fenchel dual. We leverage this duality relationship to develop new upper bounds and convergence results for the gap between the primal and dual iterates generated by these two algorithms. We also propose a new {\em primal-dual hybrid} algorithm that combines features of the conditional subgradient and mirror descent algorithms to solve the primal and dual problems in a symmetric fashion. Our algorithms and main results rely only on the availability of computable oracles for $\partial f$ and $\partial h^*$, and for $A$ and $A^*$.
We propose a combinatorial algorithm to compute the Hoffman constant of a system of linear equations and inequalities. The algorithm is based on a characterization of the Hoffman constant as the largest of a finite canonical collection of easy-to-compute Hoffman constants. Our algorithm and characterization extend to the more general context where some of the constraints are easy to satisfy as in the case of box constraints. We highlight some natural connections between our characterizations of the Hoffman constant and Renegar's distance to ill-posedness for systems of linear constraints.
This paper gives a unified and succinct approach to the $O(1/\sqrt{k}), O(1/k),$ and $O(1/k^2)$ convergence rates of the subgradient, gradient, and accelerated gradient methods for unconstrained convex minimization. In the three cases the proof of convergence follows from a generic bound defined by the convex conjugate of the objective function.
It is known that the gradient descent algorithm converges linearly when applied to a strongly convex function with Lipschitz gradient. In this case the algorithm's rate of convergence is determined by the condition number of the function. In a similar vein, it has been shown that a variant of the Frank-Wolfe algorithm with away steps converges linearly when applied to a strongly convex function with Lipschitz gradient over a polytope. In a nice extension of the unconstrained case, the algorithm's rate of convergence is determined by the product of the condition number of the function and a certain condition number of the polytope. We shed new light into the latter type of polytope conditioning. In particular, we show that previous and seemingly different approaches to define a suitable condition measure for the polytope are essentially equivalent to each other. Perhaps more interesting, they can all be unified via a parameter of the polytope that formalizes a key premise linked to the algorithm's linear convergence. We also give new insight into the linear convergence property. For a convex quadratic objective, we show that the rate of convergence is determined by a condition number of a suitably scaled polytope.
We propose a simple projection and rescaling algorithm to solve the feasibility problem \[ \text{ find } x \in L \cap Ω, \] where $L$ and $Ω$ are respectively a linear subspace and the interior of a symmetric cone in a finite-dimensional vector space $V$. This projection and rescaling algorithm is inspired by previous work on rescaled versions of the perceptron algorithm and by Chubanov's projection-based method for linear feasibility problems. As in these predecessors, each main iteration of our algorithm contains two steps: a {\em basic procedure} and a {\em rescaling} step. When $L \cap Ω\ne \emptyset$, the projection and rescaling algorithm finds a point $x \in L \cap Ω$ in at most $O(\log(1/δ(L \cap Ω)))$ iterations, where $δ(L \cap Ω) \in (0,1]$ is a measure of the most interior point in $L \cap Ω$. The ideal value $δ(L\cap Ω) = 1$ is attained when $L \cap Ω$ contains the center of the symmetric cone $Ω$. We describe several possible implementations for the basic procedure including a perceptron scheme and a smooth perceptron scheme. The perceptron scheme requires $O(r^4)$ perceptron updates and the smooth perceptron scheme requires $O(r^2)$ smooth perceptron updates, where $r$ stands for the Jordan algebra rank of $V$.
We give new insight into the Grassmann condition of the conic feasibility problem \[ x \in L \cap K \setminus\{0\}. \] Here $K\subseteq V$ is a regular convex cone and $L\subseteq V$ is a linear subspace of the finite dimensional Euclidean vector space $V$. The Grassmann condition of this problem is the reciprocal of the distance from $L$ to the set of ill-posed instances in the Grassmann manifold where $L$ lives. We consider a very general distance in the Grassmann manifold defined by two possibly different norms in $V$. We establish the equivalence between the Grassmann distance to ill-posedness of the above problem and a natural measure of the least violated trial solution to its alternative feasibility problem. We also show a tight relationship between the Grassmann and Renegar's condition measures, and between the Grassman measure and a symmetry measure of the above feasibility problem. Our approach can be readily specialized to a canonical norm in $V$ induced by $K$, a prime example being the one-norm for the non-negative orthant. For this special case we show that the Grassmann distance ill-posedness of is equivalent to a measure of the most interior solution to the above conic feasibility problem.
The von Neumann algorithm is a simple coordinate-descent algorithm to determine whether the origin belongs to a polytope generated by a finite set of points. When the origin is in the of the polytope, the algorithm generates a sequence of points in the polytope that converges linearly to zero. The algorithm's rate of convergence depends on the radius of the largest ball around the origin contained in the polytope. We show that under the weaker condition that the origin is in the polytope, possibly on its boundary, a variant of the von Neumann algorithm that includes generates a sequence of points in the polytope that converges linearly to zero. The new algorithm's rate of convergence depends on a certain geometric parameter of the polytope that extends the above radius but is always positive. Our linear convergence result and geometric insights also extend to a variant of the Frank-Wolfe algorithm with away steps for minimizing a strongly convex function over a polytope.
We show that a combination of two simple preprocessing steps would generally improve the conditioning of a homogeneous system of linear inequalities. Our approach is based on a comparison among three different but related notions of conditioning for linear inequalities.