Tamás Terlaky

Lehigh University · Department of Industrial and Systems Engineering

Quantum linear system algorithms (QLSAs) have the potential to speed up algorithms that rely on solving linear systems. Interior point methods (IPMs) yield a fundamental family of polynomial-time algorithms for solving optimization problems. IPMs solve a Newton linear system at each iteration to compute the search direction; thus, QLSAs can potenti...

The numerical performance of algorithms can be studied using test sets or procedures that generate such problems. This paper proposes various methods for generating linear, semidefinite, and second-order cone optimization problems. Specifically, we are interested in problem instances requiring a known optimal solution, a known optimal partition, a...

Second-order conic optimization (SOCO) can be considered as a special case of semidefinite optimization (SDO). In the literature it has been advised that a SOCO problem can be embedded in an SDO problem using the arrow-head matrix transformation. However, a primal-dual solution pair cannot be mapped simultaneously using the arrow-head transformatio...

Quantum linear system algorithms (QLSAs) have the potential to speed up algorithms that rely on solving linear systems. Interior Point Methods (IPMs) yield a fundamental family of polynomial-time algorithms for solving optimization problems. IPMs solve a Newton linear system at each iteration to find the search direction, and thus QLSAs can potenti...

Recent works on quantum algorithms for solving semidefinite optimization (SDO) problems have leveraged a quantum-mechanical interpretation of positive semidefinite matrices to develop methods that obtain quantum speedups with respect to the dimension $n$ and number of constraints $m$. While their dependence on other parameters suggests no overall s...

Quantum computing has the potential to speed up machine learning methods. One major direction is using quantum linear algebra to solve linear system problems or optimization problems in the machine learning area. Quantum approaches in the literature for different types of least squares problems demonstrate speedups w.r.t. dimension but have disadva...

Various noise models have been developed in quantum computing study to describe the propagation and effect of the noise which is caused by imperfect implementation of hardware. Identifying parameters such as gate and readout error rates are critical to these models. We use a Bayesian inference approach to identify posterior distributions of these p...

Discrete multi-load truss sizing optimization (MTSO) problems are challenging to solve due to their combinatorial, nonlinear, and non-convex nature. This study highlights two important characteristics of the feasible set of MTSO problems considered here, in which force balance equations, Hooke’s law, yield stress, bound constraints on displacements...

This brief note presents a personal recollection of the early history of EUROpt, the Continuous Optimization Working Group of EURO. This historical note details the events that happened before the formation of EUROpt Working Group and the first five years of its existence. During the early years EUROpt Working Group established a conference series,...

This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction. We review the notions of invariancy set, nonlinearity interval, and transition point of the optimal partition, and we investigate their characterizations. We show that the set of trans...

Quantum computing has attracted significant interest in the optimization community because it potentially can solve classes of optimization problems faster than conventional supercomputers. Several researchers proposed quantum computing methods, especially Quantum Interior Point Methods (QIPMs), to solve convex optimization problems, such as Linear...

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...

Recent claims on "solving'' combinatorial optimization problems via quantum computers have attracted many researchers to work on quantum algorithms. The max k-cut problem is a challenging combinatorial optimization problem with multiple notorious mixed integer linear optimization (MILO) formulations. Motivated by the recent progress of classical so...

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 repres...

This article explores search strategies for the design of parameterized quantum circuits. We propose several optimization approaches including random search plus survival of the fittest, reinforcement learning both with classical and hybrid quantum classical controllers, and Bayesian optimization as decision makers to design a quantum circuit in an...

Discrete variable topology optimization problems are usually solved by using solid isotropic material with penalization (SIMP), genetic algorithms (GA), or mixed-integer nonlinear optimization (MINLO). In this paper, we propose formulating discrete ply-angle and thickness topology optimization problems as a mixed-integer second-order cone optimizat...

In this paper, using an optimal partition approach, we study the parametric analysis of a second-order conic optimization problem, where the objective function is perturbed along a fixed direction. We characterize the notions of so-called invariancy set and nonlinearity interval, which serve as stability regions of the optimal partition. We then pr...

Modeling and Optimization: Theory and Applications (MOPTA) 2019 - Selected Works This special issue features a selection of contributions that were presented at the Modeling and Optimization: Theory and Applications Conference (MOPTA) held at Lehigh University in Bethlehem, Pennsylvania, USA between August 14 and August 16, 2019. This is the secon...

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...

Kinematic stability is an often overlooked, but crucial, aspect when mathematical optimization models are developed for truss topology design and sizing optimization (TTDSO) problems. In this paper, we propose a novel mixed integer linear optimization (MILO) model for the TTDSO problem with discrete cross-sectional areas and Euler buckling constrai...

This article explores search strategies for the design of parameterized quantum circuits. We propose several optimization approaches including random search plus survival of the fittest, reinforcement learning and Bayesian optimization as decision makers to design a quantum circuit in an automated way for a specific task such as multi-labeled class...

Various noise models have been developed in quantum computing study to describe the propagation and effect of the noise from imperfect implementation of hardware. In these models, critical parameters, e.g., error rate of a gate, are typically modeled as constants. Instead, we model such parameters as random variables, and apply a new Bayesian infer...

Recently, Peña and Soheili presented a deterministic rescaling perceptron algorithm and proved that it solves a feasible perceptron problem in O(m2n2 log (ρ−1)) perceptron update steps, where ρ is the radius of the largest inscribed ball. The original non-deterministic rescaling perceptron algorithm of Dunagan and Vempala is based on systematic inc...

In this paper, we study parametric analysis of semidefinite optimization problems w.r.t. the perturbation of the objective function. We study the behaviour of the optimal partition and optimal set mapping on a so-called nonlinearity interval. Furthermore, we investigate the sensitivity of the approximation of the optimal partition in a nonlinearity...

In this paper, using an optimal partition approach, we study the parametric analysis of a second-order conic optimization problem, where the objective function is perturbed along a fixed direction. We introduce the notions of nonlinearity interval and transition point of the optimal partition, and we prove that the set of transition points is finit...

This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction. We review the notions of invariancy set, nonlinearity interval, and transition point of the optimal partition, and we investigate their characterizations. We show that the continuity o...

The concept of the optimal partition was originally introduced for linear optimization and linear complementarity problems and subsequently extended to semidefinite optimization. For linear optimization and sufficient linear complementarity problems, from a central solution sufficiently close to the optimal set, the optimal partition and a maximall...

The Inmate Transportation Problem (ITP) is a common complex problem in any correctional system. We develop a weighted multi-objective mixed integer linear optimization (MILO) model for the ITP. The MILO model optimizes the transportation of the inmates within a correctional system, while considering all legal restrictions and best business practice...

It is an open question whether there is an interior-point algorithm for linear optimization problems with a lower iteration-complexity than the classical bound \(\mathcal {O}(\sqrt{n} \log (\frac{\mu _1}{\mu _0}))\). This paper provides a negative answer to that question for a variant of the Mizuno-Todd-Ye predictor-corrector algorithm. In fact, we...

This book features a selection of contributions that were presented at the Modeling and Optimization: Theory and Applications Conference (MOPTA) held at Lehigh University in B ethlehem, Pennsylvania, USA between August 16-18, 2017. The conference brought together a diverse group of researchers and practitioners working on both theoretical and pract...

Discrete truss sizing problems are very challenging to solve due to their combinatorial, nonlinear, non-convex nature. Consequently, truss sizing problems become unsolvable as the size of the truss grows. To address this issue, we consider various mathematical formulations for the truss design problem with the objective of minimizing weight, while...

Mohammad-Nezhad and Terlaky studied the identification of the optimal partition for semidefinite optimization. An approximation of the optimal partition was obtained from a bounded sequence of solutions on, or in a neighborhood of the central path. We use the approximation of the optimal partition in a rounding procedure to generate an approximate...

Under primal and dual nondegeneracy conditions, we establish the quadratic convergence of Newton's method to the unique optimal solution of second-order conic optimization. Only very few approaches have been proposed to remedy the failure of strict complementarity, mostly based on nonsmooth analysis of the optimality conditions. Our local convergen...

The inmate assignment project, in close collaboration with the Pennsylvania Department of Corrections (PADoC), took five years from start to successful implementation. In this project we developed the Inmate Assignment Decision Support System (IADSS), for which the primary goal is simultaneous and system-wide optimal assignment of inmates to correc...

In this paper, we study parametric analysis of semidefinite optimization problems with respect to the perturbation of objective function. We investigate the behavior of the optimal partition and optimal set mapping in a so called nonlinearity interval. Furthermore, we investigate the sensitivity of the approximation of the optimal partition, which...

The development of Disjunctive Conic Cuts (DCCs) for Mixed Integer Second Order Cone Optimization (MISOCO) problems has recently gained significant interest in the optimization community. In this paper, we explore the pathological disjunctions where disjunctive cuts do not tighten the description of the feasible set. We focus on the identification...

Mixed-integer second-order cone optimization (MISOCO) has become very popular in the last decade. Various aspects of solving these problems in Branch and Conic Cut (BCC) algorithms have been studied in the literature. This study aims to fill a gap and provide a novel way to find feasible solutions early in the BCC algorithm. Such solutions have a h...

Under strict complementarity and primal and dual nondegeneracy conditions we establish the quadratic convergence of Newton's method to the unique strictly complementary optimal solution of second-order conic optimization, when the initial point is sufficiently close to the optimal set. When strict complementarity fails but the primal and dual nonde...

In this paper, we consider the problem of minimizing an indefinite quadratic function subject to a single indefinite quadratic constraint. A key difficulty with this problem is its nonconvexity. Using Lagrange duality, we show that under a mild assumption, this problem can be solved by solving a linearly constrained convex univariate minimization p...

In this paper, we review the concept of the optimal partition and its identification for semidefinite optimization. In contrast to linear optimization and linear complementarity problem, it is impossible to identify the optimal partition of semidefinite optimization exactly. Instead, the sets of eigenvectors converging to an orthonormal bases for t...

The concept of the optimal partition was originally introduced for linear optimization and linear complementarity problems and subsequently extended to semidefinite optimization. For linear optimization and sufficient linear complementarity problems, the optimal partition and a maximally complementary optimal solution can be identified in strongly...

Interior point methods (IPM) are the most popular approaches to solve Second Order Cone Optimization (SOCO) problems, due to their theoretical polynomial complexity and practical performance. In this paper, we present a warm-start method for primal-dual IPMs to reduce the number of IPM steps needed to solve SOCO problems that appear in a Branch and...

The primal-dual Dikin-type affine scaling method was originally proposed for linear optimization and then extended to semidefinite optimization. Here, the method is generalized to symmetric conic optimization using the notion of Euclidean Jordan algebras. The method starts with an interior feasible but not necessarily centered primal-dual solution,...

In this paper, we propose a novel, simple, and unified approach to explore sufficient and necessary conditions, i.e., invariance conditions, under which four classic families of convex sets, namely, polyhedra, polyhedral cones, ellipsoids, and Lorenz cones, are invariant sets for a linear discrete or continuous dynamical system. For discrete dynami...

This volume contains a selection of contributions that were presented at the Modeling and Optimization: Theory and Applications Conference (MOPTA) held at Lehigh University in Bethlehem, Pennsylvania, USA on August 17-19, 2016. The conference brought together a diverse group of researchers and practitioners, working on both theoretical and practica...

We study the convex hull of the intersection of a disjunctive set defined by parallel hyperplanes and the feasible set of a mixed integer second order cone optimization (MISOCO) problem. We extend our prior work on disjunctive conic cuts (DCCs), which has thus far been restricted to the case in which the intersection of the hyperplanes and the feas...

The concept of optimal partition was originally introduced for linear optimization and linear complementary problems and subsequently extended to semidefinite optimization. For linear optimization and sufficient linear complementary problems, the optimal partition and a maximally complementary optimal solution can be identified in strongly polynomi...

Recently, Horváth et al. (Appl Math Comput, submitted) proposed a novel unified approach to study, i.e., invariance conditions, sufficient and necessary conditions, under which some convex sets are invariant sets for linear dynamical systems. In this paper, by utilizing analogous methodology, we generalize the results for nonlinear dynamical system...

In this paper, we consider local and uniform invariance preserving steplength thresholds on a set when a discretization method is applied to a linear or nonlinear dynamical system. For the forward or backward Euler method, the existence of local and uniform invariance preserving steplength thresholds is proved when the invariant sets are polyhedra,...

Recently, Mixed Integer Second Order Cone Optimization (MISOCO) has gained attention. This interest has been driven by the availability of efficient and mature methods to solve second order cone optimization (SOCO) problems and the wide range of applications of MISOCO. Financial optimization is an important application of MISOCO, where the variants...

The primal-dual Dikin-type affine scaling method was originally proposed for linear optimization and then extended to semidefinite optimization. Here, the method is generalized to symmetric conic optimization using the notion of Euclidean Jordan algebras. The method starts with an interior feasible but not necessarily centered primal-dual solution,...

For a long time the techniques of solving linear optimization (LP) problems improved only marginally. Fifteen years ago, however, a revolutionary discovery changed everything. A new `golden age' for optimization started, which is continuing up to the current time. What is the cause of the excitement? Techniques of linear programming formed previous...

We study the convex hull of the intersection of a convex set E and a disjunctive set. This intersection is at the core of solution techniques for Mixed Integer Convex Optimization. We prove that if there exists a cone K (resp., a cylinder C) that has the same intersection with the boundary of the disjunction as E, then the convex hull is the inters...

This volume contains a selection of contributions that were presented at the Modeling and Optimization: Theory and Applications Conference (MOPTA) held at Lehigh University in Bethlehem, Pennsylvania, USA on August 13-15, 2014. The conference brought together a diverse group of researchers and practitioners, working on both theoretical and practica...

It is an open question whether there is an interior-point algorithm for linear optimization problems with a lower iteration-complexity than the classical bound O(√ n log(µ1 µ0)). This paper provides a negative answer to that question for a variant of the Mizuno-Todd-Ye predictor-corrector algorithm. In fact, we prove that for any > 0, there is a re...

Steplength thresholds for invariance preserving of three types of discretization methods on a polyhedron are considered. For Taylor approximation type discretization methods we prove that a valid steplength threshold can be obtained by finding the first positive zeros of a finite number of polynomial functions. Further, a simple and efficient algor...

In this paper, we propose a novel, unified, general approach to investigate sufficient and necessary conditions under which four types of convex sets, polyhedra, polyhedral cones, ellipsoids and Lorenz cones, are invariant sets for a linear continuous or discrete dynamical system. In proving invariance of ellipsoids and Lorenz cones for discrete sy...

Full text can be downloaded (free) at: http://iopscience.iop.org/1742-6596/489/1/012038/ To enhance the measurements of radio-opaque cylindrical fiducial markers in low contrast x-ray and fluoroscopic images, a novel nonlinear marker enhancement filter (MEF) has been designed. It was primarily developed to assist in automatic initialization of a t...

This paper discusses the identification of the optimal partition of second order cone optimization (SOCO). By giving some condition numbers which only depend on the SOCO problem itself, we derive some bounds on the magnitude of the blocks of variables along the central path and prove that the optimal partition B,N,R, and T for SOCO problems can be...

For linear optimization (LO) problems, we consider a curvature integral first introduced by Sonnevend et al. (1991). Our main result states that in order to establish an upper bound for the total Sonnevend curvature of the central path, it is sufficient to consider only the case when n = 2m. This also implies that the worst cases of LO problems for...

We investigate families of quadrics all of which have the same intersection with two given hyperplanes. The cases when the two hyperplanes are parallel and when they are nonparallel are discussed. We show that these families can be described with only one parameter and describe how the quadrics are transformed as the parameter changes. This researc...

The Faculty of Information Technology and the Veszprém Regional Committee of the Hungarian Academy of Sciences (MTA-VEAB) jointly organize the Veszprém Optimization Conference: Advanced Algorithms (VOCAL) conference series. The VOCAL 2010 conference highlighted notable advances in the broadly defined area of optimization algorithms: continuous and...

This short discussion paper comments on F. Santos’ excellent review paper [ibid. 21, No. 3, 426–460 (2013; Zbl 1280.52011)]. We provide some notes on a few closely related topics that were not covered in the survey paper, such as: some relaxed version of the Hirsch conjecture; computability depending on problem data; and continuous versions of the...

Algorithms are presented for evaluating gradients and Hessians of logarithmic barrier functions for two types of convex cones: the cone of positive semidefinite matrices with a given sparsity pattern and its dual cone, the cone of sparse matrices with ...

The perceptron and the von Neumann algorithms were developed to solve linear feasibility problems. In this paper, we investigate and reveal the duality relationship between these two algorithms. The specific forms of linear feasibility problems solved by the perceptron and the von Neumann algorithms are a pair of alternative systems by the Farkas L...

The following question arises in stochastic programming: how can one approximate a noisy convex function with a convex quadratic function that is optimal in some sense. Using several approaches for constructing convex approximations we present some optimization models yielding convex quadratic regressions that are optimal approximations in L 1, L ∞...

In this chapter we highlight the relationships between multiobjective optimization and parametric optimization that is used to solve such problems. Solution of a multiobjective problem is the set of Pareto efficient points, known in the literature as Pareto efficient frontier or Pareto front. Pareto points can be obtained by using either weighting...

We consider the logarithmic and the volumetric barrier functions used in interior point methods. In the case of the logarithmic barrier function, the analytic center of a level set is the point at which the central path intersects that level set. We prove that this also holds for the volumetric path. For the central path, it is also true that the a...

This volume contains a selection of contributions that were presented at the Modeling and Optimization: Theory and Applications Conference (MOPTA) held at Lehigh University in Bethlehem, Pennsylvania, USA on August 18-20, 2010. The conference brought together a diverse group of researchers and practitioners, working on both theoretical and practica...

Global routing in VLSI (very large scale integration) design is one of the most challenging discrete optimization problems in computational theory and practice. In this paper, we present a polynomial time algorithm for the global routing problem based on an integer programming formulation. The algorithm features a theoretical approximation bound wh...

Semidefinite optimization has an ever growing family of crucial applications, and large neighborhood interior point methods (IPMs) yield the method of choice to solve them. This chapter reviews the fundamental concepts and complexity results of Self-Regular (SR) IPMs for semidefinite optimizaion, that up to a log factor achieve the best polynomial...