Research

3-elliptope
  • Conic Optimization
  • Real Algebraic Geometry
  • Perturbation and Stability Analysis
  • Convex Optimization

In preparation: 

  • Basu S, Mohammad-Nezhad A. On the central path of semidefinite optimization: degree and convergence rate.
  • Cojocaru M., Mohammad-Nezhad A., Terlaky T. On the existence of continuous selection for parametric symmetric conic optimization

Preprints:

  • Mohammad-Nezhad A., Terlaky T. On parametric second-order conic optimization (2019). arXiv:1910.03684
    • We introduced the notions of nonlinearity interval and transition point of the optimal partition for parametric second-order conic optimization. Under the strict complementarity condition, we proposed an iterative procedure to compute a nonlinearity interval of the optimal partition. Furthermore, under primal and dual nondegeneracy conditions, we showed that a transition point can be numerically identified from the higher-order derivatives of the Lagrange multipliers associated with a nonlinear reformulation of the problem. 

  • On computing the nonlinearity interval in parametric semidefinite optimization. arXiv:1908.10499
    • We studied transition points and nonlinearity intervals of a parametric semidefinite optimization, where the objective function is perturbed along a fixed direction. We described singular points of the Jacobian as an algebraic subset of R. Based on our theoretical development, we invoked procedures from numerical algebraic geometry to identify the transition points as a subset of singular points.  

Publications: 

  • Mohammad-Nezhad A., Terlaky T. Parametric analysis of semidefinite optimization. Optimization (2020) 69(1): 187-216, Click here
    • We introduced and characterized nonlinearity intervals of a parametric semidefinite optimization problem, where the objective function is perturbed along a fixed direction. The nonlinearity intervals are induced by the concept of the optimal partition, and they can be construed as stability regions where the maximal rank of primal and dual optimal solutions stay constant.
  • Mohammad-Nezhad A., Terlaky T. Quadratic convergence to the optimal solution of second-order conic optimization without strict complementarity. Optimization Methods and Software (2019) 34(5): 96-990, Click here
    • Under primal and dual nondegeneracy conditions and using the notion of the optimal partition, we established the quadratic convergence to a unique optimal solution of second-order conic optimization problem which fails the strict complementarity condition. We provide a theoretical complexity bound for identifying the quadratic convergence region of Newton’s method from the trajectory of central solutions and the so-called degree of singularity of the optimal set.
  • Mohammad-Nezhad A., Terlaky T. On the identification of optimal partition for semidefinite optimization. INFOR: Information Systems and Operational Research (2019), DOI: 10.1080/03155986.2019.1572853, Click here
    • Our goal was to approximate the optimal partition of semidefinite optimization using a bounded sequence of interior solutions, on or in a neighborhood of the central path. We derived bounds on the magnitude of the positive eigenvalues and on the convergence rate of the vanishing eigenvalues of interior solutions. Then we showed that the span of the corresponding eigenvectors give rise to an approximation of the subspaces of the optimal partition.
  • Mohammad-Nezhad A., Terlaky T. A rounding procedure for semidefinite optimization. Operations Research Letters (2019) 47:59-65, Click here
    • We proposed a methodology to generate an approximate maximally complementary optimal solution for semidefinite optimization. Using an approximation of the optimal partition, we generate a primal-dual solution with zero duality gap and approximate primal-dual feasibility w.r.t. the equality constraints, through solving two least square problems. Our rounding procedure can guess the optimal partition and then verify if the optimal partition is correct. We prove that if the rounded solution is sufficiently close to the optimal set, then it satisfies the cone constraints.
  • Shahabsafa M., Mohammad-Nezhad A., Terlaky T., Zuluaga L., He S., Hwang J., Martins J. A novel approach to discrete truss design problems using mixed integer neighborhood search. Structural and Multidisciplinary Optimization (2018) 58:2411:2429, Click here
    • We considered various mathematical formulations for the truss design problem with the objective of minimizing weight, while the cross-sectional areas of the bars take only discrete values. We proposed mixed integer linear optimization reformulations of the non-convex mixed integer models.

  • Mohammad-Nezhad A., Terlaky T. A polynomial primal-dual affine scaling algorithm for symmetric conic optimization. Computational Optimization and Applications (2016). doi:10.1007/s10589-016-9874-5 Click here
    • We generalized the so-called Dikin-type affine scaling method from linear optimization to symmetric conic optimization using Euclidean Jordan algebraic tools. A Euclidean Jordan algebra provides the machinery for defining characteristic polynomial and eigenvalues for a symmetric cone. We used a Euclidean Jordan algebra to generalize Dikin-type search directions and Dikin ellipsoid to symmetric conic optimization.

  • Mohammad Nezhad A., Mahlooji H. An artificial neural network meta-model for constrained simulation optimization. Journal of the Operational Research Society 2014;65(8): 1232-1244. Click here
  • Mohammad Nezhad A., Manzour H, Salhi S. Lagrangian relaxation heuristics for the uncapacitated single-source multi-product facility location problem. International Journal of Production Economics 2013;145(2):714-24. Click here
  • Mohammad Nezhad A. , Aliakbari Shandiz R, Eshraghniaye Jahromi A H. A particle swarm-BFGS algorithm for nonlinear programming problems. Computers and Operations Research 2013;40(4): 963-72. Click here
  • Mohammad Nezhad A. , Mahlooji H. A revised particle swarm optimization based discrete Lagrange multipliers method for nonlinear programming problems. Computers and Operations Research 2011;38(8):116474. Click here