A Vectorized Regularization Method for Multivalued Parameters Identification

2019 ◽  
Vol 16 (07) ◽  
pp. 1850110 ◽  
Author(s):  
Abdellatif Ellabib ◽  
Youssef Ouakrim
Keyword(s):  
Minimization Problem ◽  
Regularization Method ◽  
Dual Problem ◽  
Conjugate Functions ◽  
Primal Problem ◽  
Dispersion Tensor ◽  
Qualification Conditions ◽  
Primal Dual ◽  
Constraint Minimization ◽  
Stability Of The Solution

The identification of multivalued parameters is formulated as a constraint minimization problem called primal problem. We embed it in a family of perturbed problems and we associate a dual problem with it using the conjugate functions. Basing on the primal-dual relationship, under some qualification conditions on the parameters to be identified we elaborate the well posedeness, convergence and stability of the solution assuming. Numerical simulations are described in the end for the identification of discontinuous dispersion tensor in transport equations.

scholarly journals ROBUST EXPONENTIAL HEDGING AND INDIFFERENCE VALUATION

2010 ◽  
Vol 13 (07) ◽  
pp. 1075-1101 ◽  
Author(s):  
KEITA OWARI
Keyword(s):  
Dual Problem ◽  
Maximization Problem ◽  
Probability Measures ◽  
Primal Problem ◽  
Robust Version ◽  
Indifference Valuation ◽  
Robust Utility Maximization ◽  
Indifference Price ◽  
Utility Indifference ◽  
Utility Maximization Problem

We discuss the problem of exponential hedging in the presence of model uncertainty expressed by a set of probability measures. This is a robust utility maximization problem with a contingent claim. We first consider the dual problem which is the minimization of penalized relative entropy over a product set of probability measures, showing the existence and variational characterizations of the solution. These results are applied to the primal problem. Then we consider the robust version of exponential utility indifference valuation, giving the representation of indifference price using a duality result.

scholarly journals An Improved Subgradiend Optimization Technique for Solving IPs with Lagrangean Relaxation

2013 ◽  
Vol 61 (2) ◽  
pp. 135-140
Author(s):  
M Babul Hasan ◽  
Md Toha
Keyword(s):  
Dual Problem ◽  
Optimization Problems ◽  
Optimization Technique ◽  
Concave Function ◽  
Optimization Method ◽  
Duality Gap ◽  
Dual Function ◽  
Subgradient Optimization ◽  
Step Size ◽  
Primal Problem

The objective of this paper is to improve the subgradient optimization method which is used to solve non-differentiable optimization problems in the Lagrangian dual problem. One of the main drawbacks of the subgradient method is the tuning process to determine the sequence of step-lengths to update successive iterates. In this paper, we propose a modified subgradient optimization method with various step size rules to compute a tuning- free subgradient step-length that is geometrically motivated and algebraically deduced. It is well known that the dual function is a concave function over its domain (regardless of the structure of the cost and constraints of the primal problem), but not necessarily differentiable. We solve the dual problem whenever it is easier to solve than the primal problem with no duality gap. However, even if there is a duality gap the solution of the dual problem provides a lower bound to the primal optimum that can be useful in combinatorial optimization. Numerical examples are illustrated to demonstrate the method. DOI: http://dx.doi.org/10.3329/dujs.v61i2.17059 Dhaka Univ. J. Sci. 61(2): 135-140, 2013 (July)

scholarly journals Diversifikasi dan Optimasi Usahatani Terintegrasi pada Lahan Kering di Desa Musi Kecamatan Gerokgak Kabupaten Buleleng

2021 ◽  
Vol 9 (2) ◽  
pp. 416
Author(s):  
WISNO WARDANA ◽  
I Wayan Budiasa ◽  
I Ketut Suamba
Keyword(s):  
Linear Programming ◽  
Dual Problem ◽  
Problem Solution ◽  
Gross Margin ◽  
Primal Problem

Tujuan penelitian adalah (1) menganalisis besarnya pendapatan aktual (gross margin) usahatani terintegrasi (2) menganalisis apakah diversifikasi usahatani pada usahatani terintegrasi lahan kering sudah optimal. Metode yang digunakan dalam menentukan sampel pada penilitian ini adalah teknik sensus sample. Teknik sampel ini menggunakan semua anggota SIMANTRI 001 sebagai sampel dengan anggota kelompok sebanyak 20 orang. Analisis pendapatan aktual yang dipergunakan adalah analisis usahatani melalui perhitungan gross margin. Analisis optimasi dan pendapatan maksimun dianalisis menggunakan metode linear programming (LP) yang diselesaikan dengan bantuan software BPLX88. Hasil penelitian menunjukkan bahwa berdasarkan hasil analisis gross margin, dengan rata-rata luas lahan kering sebesar 0,497 ha, diperoleh pendapatan aktual usahatani jagung MT-1, jagung MT-2, kacang tanah dan ternak sapi sebesar Rp. 696.326.650 per tahun. Berdasarkan hasil analisis linear programming yang dilihat dari primal problem solution menunjukkan jagung (PJG1), jagung  (PJG2), kacang tanah (PKT) dan sapi (PSAPI) yang diusahakan bersatus basic atau profitable. Hal ini menunjukkan bahwa lahan seluas 0,497 ha telah berkontribusi dalam memperoleh pendapatan maksimum sebesar Rp. 697.333.800 per tahun. Selanjutnya pada dual problem solution, semua kendala lahan per cabang usahatani dengan luas lahan masing-masing tanaman sebesar 9,95 ha telah habis terpakai, Hal ini menunjukkan bahwa kendala lahan jagung MT-1, jagung MT-2, dan kacang tanah berstatus binding atau habis terpakai tanpa ada sisa (slack). Namun sebagian kendala tidak bersifat binding hal ini terlihat pada stok tenaga kerja bulan Januari-Desember yang belum habis digunakan. Berdasarkan analisis optimasi melalui metode linear programming dengan bantuan BLPXX8 terselenggara dengan optimal, hal ini terbukti dengan pendapatan maksimum sebesar Rp. 697.334.000 artinya mengalami peningakatan pendapatan sebesar Rp.1.007.350 (0,14%), dari pendapataan aktual saat penelitiaan sebesar Rp.696.326.650.

scholarly journals A Novel Lagrangian Multiplier Update Algorithm for Short-Term Hydro-Thermal Coordination

Energies ◽  
2020 ◽  
Vol 13 (24) ◽  
pp. 6621
Author(s):  
P. M. R. Bento ◽  
S. J. P. S. Mariano ◽  
M. R. A. Calado ◽  
L. A. F. M. Ferreira
Keyword(s):  
Lagrange Multipliers ◽  
Large Scale ◽  
Dual Problem ◽  
Electrical Power ◽  
Decomposition Methods ◽  
Subgradient Methods ◽  
Lagrangian Multiplier ◽  
Step Size ◽  
Primal Problem ◽  
Require Time

The backbone of a conventional electrical power generation system relies on hydro-thermal coordination. Due to its intrinsic complex, large-scale and constrained nature, the feasibility of a direct approach is reduced. With this limitation in mind, decomposition methods, particularly Lagrangian relaxation, constitutes a consolidated choice to “simplify” the problem. Thus, translating a relaxed problem approach indirectly leads to solutions of the primal problem. In turn, the dual problem is solved iteratively, and Lagrange multipliers are updated between each iteration using subgradient methods. However, this class of methods presents a set of sensitive aspects that often require time-consuming tuning tasks or to rely on the dispatchers’ own expertise and experience. Hence, to tackle these shortcomings, a novel Lagrangian multiplier update adaptative algorithm is proposed, with the aim of automatically adjust the step-size used to update Lagrange multipliers, therefore avoiding the need to pre-select a set of parameters. A results comparison is made against two traditionally employed step-size update heuristics, using a real hydrothermal scenario derived from the Portuguese power system. The proposed adaptive algorithm managed to obtain improved performances in terms of the dual problem, thereby reducing the duality gap with the optimal primal problem.

Monge–Kantorovich Theory

2016 ◽  
Author(s):  
Alfred Galichon
Keyword(s):  
Dual Problem ◽  
General Setting ◽  
Optimal Assignment ◽  
Primal Problem ◽  
Dual Problems ◽  
Duality Result ◽  
Equilibrium Prices ◽  
Kantorovich Problem ◽  
The Invisible

This chapter states the Monge–Kantorovich problem and provides the duality result in a fairly general setting. The primal problem is interpreted as the central planner's problem of determining the optimal assignment of workers to firms, while the dual problem is interpreted as the invisible hand's problem of obtaining a system of decentralized equilibrium prices. In general, the primal problem always has a solution (which means that an optimal assignment of workers to jobs exists), but the dual does not: the optimal assignment cannot always be decentralized by a system of prices. However, the cases where the dual problem does not have a solution are rather pathological, and in all of the cases considered in the rest of the book, both the primal and the dual problems have solutions.

scholarly journals The Use of the Duality Principle to Solve Optimization Problems

2018 ◽  
Vol 6 (1) ◽  
pp. 33
Author(s):  
Rowland Jerry Okechukwu Ekeocha ◽  
Chukwunedum Uzor ◽  
Clement Anetor
Keyword(s):  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Linear Program ◽  
Duality Gap ◽  
Duality Principle ◽  
Primal Problem ◽  
Optimal Values ◽  
Primal And Dual Problems ◽  
Primal And Dual

<p><span>The duality principle provides that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. However the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition.<span>  </span>In other words given any linear program, there is another related linear program called the dual. In this paper, an understanding of the dual linear program will be developed. This understanding will give important insights into the algorithm and solution of optimization problem in linear programming. <span> </span>Thus the main concepts of duality will be explored by the solution of simple optimization problem.</span></p>

On a Noncompact Minimization Problem of Hardy-Sobolev Type

2002 ◽  
Vol 2 (1) ◽  
Author(s):  
K. Sandeep
Keyword(s):  
Minimization Problem ◽  
Sobolev Type ◽  
Existence And Nonexistence ◽  
Constraint Minimization

AbstractIn this paper we discuss the existence and nonexistence of minimizer for the constraint minimization problem

scholarly journals The back-and-forth method for Wasserstein gradient flows

2021 ◽  
Vol 27 ◽  
pp. 28
Author(s):  
Matt Jacobs ◽  
Wonjun Lee ◽  
Flavien Léger
Keyword(s):  
Large Class ◽  
Large Scale ◽  
Dual Problem ◽  
Optimal Transport ◽  
Gradient Flow ◽  
Gradient Flows ◽  
Primal Problem ◽  
Transport Problems ◽  
Wasserstein Gradient Flows ◽  
Numer Math

We present a method to efficiently compute Wasserstein gradient flows. Our approach is based on a generalization of the back-and-forth method (BFM) introduced in Jacobs and Léger [Numer. Math. 146 (2020) 513–544.]. to solve optimal transport problems. We evolve the gradient flow by solving the dual problem to the JKO scheme. In general, the dual problem is much better behaved than the primal problem. This allows us to efficiently run large scale gradient flows simulations for a large class of internal energies including singular and non-convex energies.

Duality Theory

1995 ◽  
Author(s):  
Christodoulos A. Floudas
Keyword(s):  
Duality Theory ◽  
Dual Problem ◽  
Optimization Problems ◽  
Strong Duality ◽  
Perturbation Function ◽  
Primal Problem ◽  
Theory Section ◽  
Existence Conditions ◽  
Definition Of ◽  
Optimal Multiplier

Nonlinear optimization problems have two different representations, the primal problem and the dual problem. The relation between the primal and the dual problem is provided by an elegant duality theory. This chapter presents the basics of duality theory. Section 4.1 discusses the primal problem and the perturbation function. Section 4.2 presents the dual problem. Section 4.3 discusses the weak and strong duality theorems, while section 4.4 discusses the duality gap. This section presents the formulation of the primal problem, the definition and properties of the perturbation function, the definition of stable primal problem, and the existence conditions of optimal multiplier vectors.

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