scholarly journals Interior-point algorithms for a class of convex optimization problems

2009 ◽  
Vol 19 (2) ◽  
pp. 239-248 ◽  
Author(s):  
Goran Lesaja ◽  
Verlynda Slaughter
Keyword(s):  
Convex Optimization ◽  
Interior Point ◽  
Original Problem ◽  
Optimization Problems ◽  
Linear Constraints ◽  
Quadratic Problem ◽  
Convex Optimization Problem ◽  
Convex Optimization Problems ◽  
Novel Approach ◽  
Better Than

In this paper we consider interior-point methods (IPM) for the nonlinear, convex optimization problem where the objective function is a weighted sum of reciprocals of variables subject to linear constraints (SOR). This problem appears often in various applications such as statistical stratified sampling and entropy problems, to mention just few examples. The SOR is solved using two IPMs. First, a homogeneous IPM is used to solve the Karush-Kuhn-Tucker conditions of the problem which is a standard approach. Second, a homogeneous conic quadratic IPM is used to solve the SOR as a reformulated conic quadratic problem. As far as we are aware of it, this is a novel approach not yet considered in the literature. The two approaches are then numerically tested on a set of randomly generated problems using optimization software MOSEK. They are compared by CPU time and the number of iterations, showing that the second approach works better for problems with higher dimensions. The main reason is that although the first approach increases the number of variables, the IPM exploits the structure of the conic quadratic reformulation much better than the structure of the original problem.

scholarly journals Novel Interior Point Algorithms for Solving Nonlinear Convex Optimization Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Sakineh Tahmasebzadeh ◽  
Hamidreza Navidi ◽  
Alaeddin Malek
Keyword(s):  
Convex Optimization ◽  
Objective Function ◽  
Interior Point ◽  
Optimization Problems ◽  
Numerical Algorithms ◽  
Optimal Solution ◽  
Linear Constraints ◽  
Convex Optimization Problems ◽  
Novel Algorithms ◽  
Interior Point Technique

This paper proposes three numerical algorithms based on Karmarkar’s interior point technique for solving nonlinear convex programming problems subject to linear constraints. The first algorithm uses the Karmarkar idea and linearization of the objective function. The second and third algorithms are modification of the first algorithm using the Schrijver and Malek-Naseri approaches, respectively. These three novel schemes are tested against the algorithm of Kebiche-Keraghel-Yassine (KKY). It is shown that these three novel algorithms are more efficient and converge to the correct optimal solution, while the KKY algorithm fails in some cases. Numerical results are given to illustrate the performance of the proposed algorithms.

scholarly journals Multivariate Spectral Gradient Algorithm for Nonsmooth Convex Optimization Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Yaping Hu
Keyword(s):  
Convex Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Gradient Algorithm ◽  
Nonsmooth Convex Optimization ◽  
Convex Optimization Problem ◽  
Convex Optimization Problems ◽  
Yosida Regularization ◽  
Approximate Function ◽  
Original Objective

We propose an extended multivariate spectral gradient algorithm to solve the nonsmooth convex optimization problem. First, by using Moreau-Yosida regularization, we convert the original objective function to a continuously differentiable function; then we use approximate function and gradient values of the Moreau-Yosida regularization to substitute the corresponding exact values in the algorithm. The global convergence is proved under suitable assumptions. Numerical experiments are presented to show the effectiveness of this algorithm.

Counterweight Balancing for Reducing Machine Frame Vibration: A Convex Optimization Framework

2006 ◽  
Author(s):  
Myriam Verschuure ◽  
Bram Demeulenaere ◽  
Jan Swevers ◽  
Joris De Schutter
Keyword(s):  
Convex Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Global Optimum ◽  
Convex Optimization Problem ◽  
Optimization Framework ◽  
Convex Optimization Problems ◽  
Drive Speed ◽  
Maximal Reduction

This paper focusses on reducing, through counterweight addition, the vibration of an elastically mounted, rigid machine frame that supports a linkage. In order to determine the counterweights that yield a maximal reduction in frame vibration, a non-linear optimization problem is formulated with the frame kinetic energy as objective function and such that a convex optimization problem is obtained. Convex optimization problems are nonlinear optimization problems that have a unique (global) optimum, which can be found with great efficiency. The proposed methodology is successfully applied to improve the results of the benchmark four-bar problem, first considered by Kochev and Gurdev. For this example, the balancing is shown to be very robust for drive speed variations and to benefit only marginally from using a coupler counterweight.

scholarly journals New inertial proximal gradient methods for unconstrained convex optimization problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Peichao Duan ◽  
Yiqun Zhang ◽  
Qinxiong Bu
Keyword(s):  
Convex Optimization ◽  
Optimization Problems ◽  
Gradient Methods ◽  
Gradient Algorithm ◽  
Proximal Gradient Method ◽  
Proximal Point ◽  
Convex Optimization Problem ◽  
Convex Optimization Problems ◽  
Acceleration Methods ◽  
Weak Convergence Theorem

AbstractThe proximal gradient method is a highly powerful tool for solving the composite convex optimization problem. In this paper, firstly, we propose inexact inertial acceleration methods based on the viscosity approximation and proximal scaled gradient algorithm to accelerate the convergence of the algorithm. Under reasonable parameters, we prove that our algorithms strongly converge to some solution of the problem, which is the unique solution of a variational inequality problem. Secondly, we propose an inexact alternated inertial proximal point algorithm. Under suitable conditions, the weak convergence theorem is proved. Finally, numerical results illustrate the performances of our algorithms and present a comparison with related algorithms. Our results improve and extend the corresponding results reported by many authors recently.

Sign in / Sign up

Export Citation Format

Share Document