arXiv subjects

Levent Tuncel

Publications and source records attributed to Levent Tuncel.

10 recordsLinked to original sources

arXiv2026

Efficient Interior-Point Methods for Hyperbolic Programming via Straight-Line Programs

Hyperbolic (HB) programming generalizes many popular convex optimization problems, including semidefinite and second-order cone programming. Despite substantial theoretical progress on HB programming, efficient computational tools for solving large-scale hyperbolic programs remain limited. This paper presents DDS 3.0, a new release of the Domain-Driven Solver, which provides an efficient interior-point implementation tailored for hyperbolic programming. A key innovation lies in a new straight-line program (SLP) representation that enables compact representation and efficient computation of hyperbolic polynomials, their gradients, and Hessians. The SLP structure significantly reduces computational cost, allowing the Hessian to be computed in the same asymptotic complexity as the gradient through a batched reverse-over-forward differentiation scheme. We further introduce an improved corrector step for the primal-dual interior-point method, enhancing stability and convergence on convex sets where only the primal self-concordant barrier is efficiently computable. We create a comprehensive benchmark library beyond the elementary symmetric polynomials, using several different techniques. Numerical experiments demonstrate substantial performance gains of DDS 3.0 compared to first-order Frank-Wolfe algorithm, homotopy method, and SDP software utilizing SDP relaxations.

math.OC↗

arXiv2023

Efficient Implementation of Interior-Point Methods for Quantum Relative Entropy

Quantum Relative Entropy (QRE) programming is a recently popular and challenging class of convex optimization problems with significant applications in quantum computing and quantum information theory. We are interested in modern interior point (IP) methods based on optimal self-concordant barriers for the QRE cone. A range of theoretical and numerical challenges associated with such barrier functions and the QRE cones have hindered the scalability of IP methods. To address these challenges, we propose a series of numerical and linear algebraic techniques and heuristics aimed at enhancing the efficiency of gradient and Hessian computations for the self-concordant barrier function, solving linear systems, and performing matrix-vector products. We also introduce and deliberate about some interesting concepts related to QRE such as symmetric quantum relative entropy (SQRE). We also introduce a two-phase method for performing facial reduction that can significantly improve the performance of QRE programming. Our new techniques have been implemented in the latest version (DDS 2.2) of the software package DDS. In addition to handling QRE constraints, DDS accepts any combination of several other conic and non-conic convex constraints. Our comprehensive numerical experiments encompass several parts including 1) a comparison of DDS 2.2 with Hypatia for the nearest correlation matrix problem, 2) using DDS for combining QRE constraints with various other constraint types, and 3) calculating the key rate for quantum key distribution (QKD) channels and presenting results for several QKD protocols.

quant-ph↗

arXiv2017

A Utility Theory Based Interactive Approach to Robustness in Linear Optimization

We treat uncertain linear programming problems by utilizing the notion of weighted analytic centers and notions from the area of multi-criteria decision making. After introducing our approach, we develop interactive cutting-plane algorithms for robust optimization, based on concave and quasi-concave utility functions. In addition to practical advantages, due to the flexibility of our approach, we are able to prove that under a theoretical framework due to Bertsimas and Sim [14], which establishes the existence of certain convex formulation of robust optimization problems, the robust optimal solutions generated by our algorithms are at least as desirable to the decision maker as any solution generated by many other robust optimization algorithms in the theoretical framework. We present some probabilistic bounds for feasibility of robust solutions and evaluate our approach by means of computational experiments.

math.OC↗

arXiv2015

Approximation Ratio of LD Algorithm for Multi-Processor Scheduling and the Coffman-Sethi Conjecture

Coffman and Sethi proposed a heuristic algorithm, called LD, for multi-processor scheduling, to minimize makespan over flowtime-optimal schedules. LD algorithm is a natural extension of a very well-known list scheduling algorithm, Longest Processing Time (LPT) list scheduling, to our bicriteria scheduling problem. Moreover, in 1976, Coffman and Sethi conjectured that LD algorithm has precisely the following worst-case performance bound: $\frac{5}{4} - \frac{3}{4(4m-1)}$, where m is the number of machines. In this paper, utilizing some recent work by the authors and Huang, from 2013, which exposed some very strong combinatorial properties of various presumed minimal counterexamples to the conjecture, we provide a proof of this conjecture. The problem and the LD algorithm have connections to other fundamental problems (such as the assembly line-balancing problem) and to other algorithms.

cs.DS↗

arXiv2014

Local Superlinear Convergence of Polynomial-Time Interior-Point Methods for Hyperbolic Cone Optimization Problems

In this paper, we establish the local superlinear convergence property of some polynomial-time interior-point methods for an important family of conic optimization problems. The main structural property used in our analysis is the logarithmic homogeneity of self-concordant barrier function, which must have {\em negative curvature}. We propose a new path-following predictor-corrector scheme, which work only in the dual space. It is based on an easily computable gradient proximity measure, which ensures an automatic transformation of the global linear rate of convergence to the local superlinear one under some mild assumptions. Our step-size procedure for the predictor step is related to the maximum step size maintaining feasibility. As the optimal solution set is approached, our algorithm automatically tightens the neighborhood of the central path proportionally to the current duality gap.

math.OC↗

arXiv2014

A Perturbed Sums of Squares Theorem for Polynomial Optimization and its Applications

We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by $(f^\ast - ε)$ and from above by $f^\ast + ε(n+1)$, where $f^\ast$ is the optimal value of the POP. We propose new SDP relaxations for POP based on modifications of existing sums-of-squares representation theorems. An advantage of our SDP relaxations is that in many cases they are of considerably smaller dimension than those originally proposed by Lasserre. We present some applications and the results of our computational experiments.

math.OC↗

arXiv2013

Worst-Case Performance Analysis of Some Approximation Algorithms for Minimizing Makespan and Flow-Time

In 1976, Coffman and Sethi conjectured that a natural extension of LPT list scheduling to the bicriteria scheduling problem of minimizing makespan over flowtime optimal schedules, called LD algorithm, has a simple worst-case performance bound: (5m-2)/(4m-1), where m is the number of machines. We study structure of potential minimal counterexamples to this conjecture and prove that the conjecture holds for the cases (i) n > 5m, (ii) m = 2, (iii) m = 3, and (iv) m greater than or equal to 4, n less than or equal to 3m, where n is the number of jobs. We further conclude that to verify the conjecture, it suffices to analyze the following case: for every m greater than or equal to 4, n is either equal to 4m or 5m.

cs.DS↗

arXiv2000

When Does the Positive Semidefiniteness Constraint Help In Lifting Procedures

We study the lift-and-project procedures of Lovász and Schrijver for 0-1 integer programming problems. We prove that the procedure using the positive semidefiniteness constraint is not better than the one without it, in the worst case. Various examples are considered. We also provide geometric conditions characterizing when the positive semidefiniteness constraint does not help.

math.CO↗

arXiv2000

Some Fundamental Properties of Successive Convex Relaxation Methods on LCP and Related Problems

General Successive Convex Relaxation Methods (SRCMs) can be used to compute the convex hull of any compact set, in an Euclidean space, described by a system of quadratic inequalities and a compact convex set which is not very complicated. Linear Complementarity Problems (LCPs) make an interesting and rich class of structured nonconvex optimization problems. In this paper, we study a few of the specialized lift-and-project methods and some of the possible ways of applying the general SCRMs to LCPs and related problems.

math.CO↗

arXiv1999

Some fundamental properties of successive convex relaxation methods on LCP and related problems

math.OC↗