The Primal-Dual Model of an Objective Function

2018 ◽  
pp. 423-487
Author(s):  
Yurii Nesterov
Keyword(s):  
Objective Function ◽  
Dual Model ◽  
Primal Dual

scholarly journals Nondifferentiable G-Mond–Weir Type Multiobjective Symmetric Fractional Problem and Their Duality Theorems under Generalized Assumptions

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1348 ◽  
Author(s):  
Ramu Dubey ◽  
Lakshmi Narayan Mishra ◽  
Luis Manuel Sánchez Ruiz
Keyword(s):  
Vector Optimization ◽  
Optimization Problem ◽  
Vector Optimization Problem ◽  
Second Order ◽  
Type Model ◽  
Dual Model ◽  
Duality Theorems ◽  
Converse Duality ◽  
Numerical Example ◽  
Primal Dual

In this article, a pair of nondifferentiable second-order symmetric fractional primal-dual model (G-Mond–Weir type model) in vector optimization problem is formulated over arbitrary cones. In addition, we construct a nontrivial numerical example, which helps to understand the existence of such type of functions. Finally, we prove weak, strong and converse duality theorems under aforesaid assumptions.

scholarly journals Nonlinear Least Square Based on Control Direction by Dual Method and Its Application

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Zhengqing Fu ◽  
Guolin Liu ◽  
Lanlan Guo ◽  
Weike Liu ◽  
Hua Guo
Keyword(s):  
Quantitative Analysis ◽  
Root Mean Square Error ◽  
Qualitative Analysis ◽  
Mean Square Error ◽  
Root Mean Square ◽  
Least Square ◽  
Dual Method ◽  
Dual Model ◽  
Mean Square ◽  
Primal Dual

A direction controlled nonlinear least square (NLS) estimation algorithm using the primal-dual method is proposed. The least square model is transformed into the primal-dual model; then direction of iteration can be controlled by duality. The iterative algorithm is designed. The Hilbert morbid matrix is processed by the new model and the least square estimate and ridge estimate. The main research method is to combine qualitative analysis and quantitative analysis. The deviation between estimated values and the true value and the estimated residuals fluctuation of different methods are used for qualitative analysis. The root mean square error (RMSE) is used for quantitative analysis. The results of experiment show that the model has the smallest residual error and the minimum root mean square error. The new estimate model has effectiveness and high precision. The genuine data of Jining area in unwrapping experiments are used and the comparison with other classical unwrapping algorithms is made, so better results in precision aspects can be achieved through the proposed algorithm.

A Convex and Exact Approach to Discrete Constrained TV-L1 Image Approximation

2011 ◽  
Vol 1 (2) ◽  
pp. 172-186 ◽  
Author(s):  
Jing Yuan ◽  
Juan Shi ◽  
Xue-Cheng Tai
Keyword(s):  
Optimization Problem ◽  
Convex Relaxation ◽  
Dual Model ◽  
Convex Optimization Problem ◽  
Image Approximation ◽  
Primal Dual ◽  
Image Function ◽  
Dual Perspective ◽  
Primal And Dual ◽  
Theoretical Results

AbstractWe study the TV-L1 image approximation model from primal and dual perspective, based on a proposed equivalent convex formulations. More specifically, we apply a convex TV-L1 based approach to globally solve the discrete constrained optimization problem of image approximation, where the unknown image function u(x) ∈ {f1,…,fn}, ∀x ∈ Ω. We show that the TV-L1 formulation does provide an exact convex relaxation model to the non-convex optimization problem considered. This result greatly extends recent studies of Chan et al., from the simplest binary constrained case to the general gray-value constrained case, through the proposed rounding scheme. In addition, we construct a fast multiplier-based algorithm based on the proposed primal-dual model, which properly avoids variability of the concerning TV-L1 energy function. Numerical experiments validate the theoretical results and show that the proposed algorithm is reliable and effective.

scholarly journals A primal-dual approach of weak vector equilibrium problems

2018 ◽  
Vol 16 (1) ◽  
pp. 276-288 ◽  
Author(s):  
Szilárd László
Keyword(s):  
Original Problem ◽  
Equilibrium Problems ◽  
Sufficient Conditions ◽  
Dual Model ◽  
Vector Equilibrium Problems ◽  
Dual Approach ◽  
Topological Vector Spaces ◽  
Primal Dual ◽  
Vector Equilibrium ◽  
Weak Vector

AbstractIn this paper we provide some new sufficient conditions that ensure the existence of the solution of a weak vector equilibrium problem in Hausdorff topological vector spaces ordered by a cone. Further, we introduce a dual problem and we provide conditions that assure the solution set of the original problem and its dual coincide. We show that many known problems from the literature can be treated in our primal-dual model. We provide several coercivity conditions in order to obtain the existence of the solution of the primal-dual problems without compactness assumption. We apply the obtained results to perturbed vector equilibrium problems.

scholarly journals Optimization In A Finite-Dimensional Euclide Space

2020 ◽  
Vol 7 (1) ◽  
pp. 20-28
Author(s):  
A.I. Kosolap ◽  
Keyword(s):  
Euclidean Space ◽  
Objective Function ◽  
Optimization Model ◽  
Optimization Problems ◽  
Complexity Class ◽  
Feasible Region ◽  
Complexity Classes ◽  
The Third ◽  
Primal Dual ◽  
Multimodal Problems

In this paper, optimization models in Euclidean space are divided into four complexity classes. Ef-fective algorithms have been developed to solve the problems of the first two classes of complexity. These are the primal-dual interior-point methods. Discrete and combinatorial optimization problems of the third complexity class are recommended to be converted to the fourth complexity class with continuous change of variables. Effective algorithms have not been developed for problems of the third and fourth complexity classes, with the exception of a narrow class of problems that are unimodal. The general optimization problem is formulated as a minimum (maximum) objective function in the presence of constraints. The complexity of the problem depends on the structure of the objective function and its feasible region. If the functions that determine the optimization model are quadratic or polynomial, then semidefinite programming can be used to obtain estimates of so-lutions in such problems. Effective methods have been developed for semidefinite optimization problems. Sometimes it’s enough to develop an algorithm without building a mathematical model. We see such an example when sorting an array of numbers. Effective algorithms have been devel-oped to solve this problem. In the work for sorting problems, an optimization model is constructed, and it coincides with the model of the assignment problem. It follows from this that the sorting problem is unimodal. Effective algorithms have not been developed to solve multimodal problems. The paper proposes a simple and effective algorithm for the optimal allocation of resources in mul-tiprocessor systems. This problem is multimodal. In the general case, for solving multimodal prob-lems, a method of exact quadratic regularization is proposed. This method has proven its compara-tive effectiveness in solving many test problems of various dimensions. Keywords: Euclidean space, optimization, unimodal problems, multimodal problems, complexity classes, numerical methods.

scholarly journals A Proposed Technique for Solving Linear Fractional Bounded Variable Problems

2012 ◽  
Vol 60 (2) ◽  
pp. 223-230
Author(s):  
H.K. Das ◽  
M. Babul Hasan
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Programming Language ◽  
Simplex Algorithm ◽  
Computational Effort ◽  
New Method ◽  
Bounded Variables ◽  
Primal Dual ◽  
Dual Simplex ◽  
Dual Simplex Algorithm

In this paper, a new method is proposed for solving the problem in which the objective function is a linear fractional Bounded Variable (LFBV) function, where the constraints functions are in the form of linear inequalities and the variables are bounded. The proposed method mainly based upon the primal dual simplex algorithm. The Linear Programming Bounded Variables (LPBV) algorithm is extended to solve Linear Fractional Bounded Variables (LFBV).The advantages of LFBV algorithm are simplicity of implementation and less computational effort. We also compare our result with programming language MATHEMATICA.DOI: http://dx.doi.org/10.3329/dujs.v60i2.11522 Dhaka Univ. J. Sci. 60(2): 223-230, 2012 (July) 

Optimisasi Model Analisa Sensitivitas Primal–Dual dalam Management Produksi Dodol

2019 ◽  
Vol 2 (2) ◽  
Author(s):  
Iin Maulina ◽  
Siti Rusdiana ◽  
Rini Oktavia
Keyword(s):  
Sensitivity Analysis ◽  
Objective Function ◽  
Operation Research ◽  
Raw Materials ◽  
Applied Mathematics ◽  
Optimal Solution ◽  
Optimal Level ◽  
Research Problems ◽  
Dual Analysis ◽  
Primal Dual

Optimalisasi adalah cabang matematika terapan yang mempelajari masalah-masalah operasi riset dengan tujuan untuk memaksimalkan atau meminimalkan besaran tertentu. Dalam penelitian ini, kami akan menerapkan konsep optimasi pada mengoptimalkan kegiatan produksi di CV. Sentra Halia Sabang yang merupakan industri rumah tangga yang memproduksi dodol dalam kotak kemasan. Dengan melakukan penelitian tentang kegiatan produksi di CV. Sentra Halia Sabang, ditentukan bahwa fungsi tujuan adalah untuk memaksimalkan keuntungan perusahaan, yang dimodelkan sebagai Z=7537x1+5871x2+7887x3+6151x4+7987x5+6231x6+7909x7+6171x8+8037x9+6271x10+8629x11, di mana xi mewakili jumlah produksi untuk setiap jenis dodol. Selanjutnya, juga ditinjau sumber daya terbatas yang digunakan oleh perusahaan seperti jumlah bahan baku, waktu, maksimum dan batas minimum produksi. Pemrosesan data dilakukan dengan menggunakan perangkat lunak LINDO. Analisis yang digunakan adalah analisis dual primal yang dilanjutkan dengan analisis sensitivitas. Hasil penelitian ini menunjukkan bahwa produksi CV. Sentra Halia Sabang hampir mencapai level optimal. Jika perusahaan berproduksi dalam kondisi optimal, laba yang bisa diperoleh adalah Rp 195.330. Analisis dual primal menunjukkan bahwa ada sumber daya yang belum digunakan secara optimal, yang bisa dilihat dari nilai slack / surplus pada beberapa sumber daya. Dalam analisis sensitivitas, beberapa variasi penambahan pada masing-masing koefisien fungsi tujuan diterapkan. Hasil analisis sensitivitas menunjukkan bahwa penambahan koefisien fungsi tujuan sebesar Rp 150 tidak akan mempengaruhi solusi optimal awal.   Optimization is a part of applied mathematics that studies operation research problems with the aim to maximize or minimize a certain magnitude. In this research, we will apply the optimization concepts on optimizing production activities in CV. Sentra Halia Sabang which is a home industry producing dodol in a packing box. By doing research on production activities at CV. Sentra Halia Sabang, it was determined that the objective function is to maximize the profits of the company, which is modeled as Z=7537x1+5871x2+7887x3+6151x4+7987x5+6231x6+7909x7+6171x8+8037x9+6271x10+8629x11, where xi represents the amount of productions for each type of dodol. Next, it is also reviewed the limited resources used by the company such as the amount of raw materials, the time, the maximum and the minimum limit of production. Data processing is performed using software LINDO. The analyses applied were the primal dual analysis that was continued with the sensitivity analysis. The results of this study indicate that the production of CV. Sentra Halia Sabang almost reached the optimal level. If the company produces in optimal conditions, the profit that can be obtained is Rp 195 430. The primal dual analyses show that there are resources that have not been used optimally, that can be seen from the value of slack / surplus on some resources. In the sensitivity analysis, some various addition on each of the coefficients of the objective function were applied. The result of the sensitivity analysis show that the addition of the coefficients of the objective function as much as Rp 150 will not affect the initial optimal solution.

Sign in / Sign up

Export Citation Format

Share Document