Ali Mohammad Nezhad

University of North Carolina at Chapel Hill | UNC · Department of Statistics and Operations Research

In this paper, a new algorithm for solving constrained nonlinear programming problems is presented. The basis of our proposed algorithm is none other than the necessary and sufficient conditions that one deals within a discrete constrained local optimum in the context of the discrete Lagrange multipliers theory. We adopt a revised particle swarm op...

It is well-known that the central path of semi-definite optimization, unlike linear optimization, has no analytic extension to $\mu = 0$ in the absence of the strict complementarity condition. In this paper, we show the existence of a positive integer $\rho$ by which the reparametrization $\mu \mapsto \mu^{\rho}$ recovers the analyticity of the cen...

Let $\mathrm{R}$ be a real closed field. Given a closed and bounded semi-algebraic set $A \subset \mathrm{R}^n$ and semi-algebraic continuous functions $f,g:A \rightarrow \mathrm{R}$, such that $f^{-1}(0) \subset g^{-1}(0)$, there exist $N$ and $c \in \mathrm{R}$, such that the inequality ({\L}ojasiewicz inequality) $|g(x)|^N \le c \cdot |f(x)|$ ho...

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...

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...

In this paper, we investigate the complexity of the central path of semidefinite optimization through the lens of real algebraic geometry. To that end, we propose an algorithm to compute real univariate representations describing the central path and its limit point, where the limit point is described by taking the limit of central solutions, as bo...

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...

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...

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...

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 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...

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,...

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...

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,...

This paper presents artificial neural network (ANN) meta-models for expensive continuous simulation optimization (SO) with stochastic constraints. These meta-models are used within a sequential experimental design to approximate the objective function and the stochastic constraints. To capture the non-linear nature of the ANN, the SO problem is ite...

Facility location problem is one of the strategic logistical drivers within the supply chain which is a hard to solve optimization problem. In this study, we focus on the uncapacitated single-source multi-product production/distribution facility location problem with the presence of set-up cost. To efficiently tackle this decision problem, two la...