The optimal value and optimal solutions of the proximal average of convex functions

2012 ◽  
Vol 75 (3) ◽  
pp. 1290-1304 ◽  
Author(s):  
Rafal Goebel ◽  
Warren Hare ◽  
Xianfu Wang
Keyword(s):  
Convex Functions ◽  
Optimal Solutions ◽  
Proximal Average ◽  
Optimal Value

scholarly journals On quantitative stability in infinite-dimensional optimization under uncertainty

2021 ◽  
Author(s):  
M. Hoffhues ◽  
W. Römisch ◽  
T. M. Surowiec
Keyword(s):  
Stochastic Optimization ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Optimization Under Uncertainty ◽  
Optimal Solutions ◽  
Probability Metrics ◽  
Pde Constrained Optimization ◽  
Quantitative Stability ◽  
Infinite Dimensional ◽  
Optimal Value

AbstractThe vast majority of stochastic optimization problems require the approximation of the underlying probability measure, e.g., by sampling or using observations. It is therefore crucial to understand the dependence of the optimal value and optimal solutions on these approximations as the sample size increases or more data becomes available. Due to the weak convergence properties of sequences of probability measures, there is no guarantee that these quantities will exhibit favorable asymptotic properties. We consider a class of infinite-dimensional stochastic optimization problems inspired by recent work on PDE-constrained optimization as well as functional data analysis. For this class of problems, we provide both qualitative and quantitative stability results on the optimal value and optimal solutions. In both cases, we make use of the method of probability metrics. The optimal values are shown to be Lipschitz continuous with respect to a minimal information metric and consequently, under further regularity assumptions, with respect to certain Fortet-Mourier and Wasserstein metrics. We prove that even in the most favorable setting, the solutions are at best Hölder continuous with respect to changes in the underlying measure. The theoretical results are tested in the context of Monte Carlo approximation for a numerical example involving PDE-constrained optimization under uncertainty.

scholarly journals Greedy systems of linear inequalities and lexicographically optimal solutions

2019 ◽  
Vol 53 (5) ◽  
pp. 1929-1935
Author(s):  
Satoru Fujishige
Keyword(s):  
Convex Functions ◽  
Linear Inequalities ◽  
Convex Polyhedra ◽  
Optimal Solutions ◽  
Integral Points ◽  
System Of Linear Inequalities ◽  
Discrete Convexity ◽  
The Greedy Algorithm

The present note reveals the role of the concept of greedy system of linear inequalities played in connection with lexicographically optimal solutions on convex polyhedra and discrete convexity. The lexicographically optimal solutions on convex polyhedra represented by a greedy system of linear inequalities can be obtained by a greedy procedure, a special form of which is the greedy algorithm of J. Edmonds for polymatroids. We also examine when the lexicographically optimal solutions become integral. By means of the Fourier–Motzkin elimination Murota and Tamura have recently shown the existence of integral points in a polyhedron arising as a subdifferential of an integer-valued, integrally convex function due to Favati and Tardella [Murota and Tamura, Integrality of subgradients and biconjugates of integrally convex functions. Preprint arXiv:1806.00992v1 (2018)], which can be explained by our present result. A characterization of integrally convex functions is also given.

Statistical Inference of Second-Order Cone Programming

2018 ◽  
Vol 35 (06) ◽  
pp. 1850044
Author(s):  
Jiani Wang ◽  
Liwei Zhang
Keyword(s):  
Objective Function ◽  
Second Order ◽  
Optimal Solutions ◽  
Random Vectors ◽  
Cone Programming ◽  
Second Order Cone Programming ◽  
Second Order Cone ◽  
Analysis Theory ◽  
Optimal Value ◽  
Sum Of Norms

The randomness of the second-order cone programming problems is mainly reflected in the objective function and the constraints both having random vectors. In this paper, we discuss the statistical properties of estimates of the respective optimal value and optimal solutions when the random vectors are estimated by their sample both in the objective function and the constraints, which are based on perturbation analysis theory of second-order cone programming. As an example we consider the problem of minimizing a sum of norms with weights.

scholarly journals Modeling of complex structured processes using discrete iterative networks and petri nets

2021 ◽  
Vol 2131 (3) ◽  
pp. 032004
Author(s):  
A Korneev ◽  
T Lavrukhina ◽  
T Smetannikova ◽  
Yu Glazkova
Keyword(s):  
Petri Nets ◽  
Optimization Methods ◽  
Optimal Solutions ◽  
Processing Mode ◽  
Functional Relationships ◽  
Multi Stage ◽  
Optimal Value ◽  
Spatially Distributed ◽  
Alternative Solutions ◽  
Production Conditions

Abstract The paper considers the use of discrete iterative networks and Petri nets in the modeling of complex structured processes. In general, the production process is represented as a composition of machines. Probabilistic finite machine are modeled to select optimal solutions from a certain set of alternative solutions. Alphabets of finite and probabilistic machines are formed on the basis of discrete optimization methods when modeling multi-stage productions. The processing mode adaptation block is used to change the alphabets when the production conditions are changed. Using the alphabet generation block allows you to choose the optimal value of the alphabets of the random variables under study. During modeling complex systems and developing algorithms for managing them, the presence of ambiguous functional relationships between factors and quality indicators is taken into account. Methods of modeling complex spatially distributed objects based on a hierarchy of machines belonging to a predetermined finite number of machine types using iterative networks and Petri nets are described.

scholarly journals Pessimistic Bilevel Linear Optimization

2018 ◽  
Vol 1 (1) ◽  
pp. 1-10
Author(s):  
S. Dempe ◽  
G. Luo ◽  
S. Franke
Keyword(s):  
Optimization Problem ◽  
Value Function ◽  
Linear Optimization ◽  
Equivalent Transformation ◽  
Optimal Solutions ◽  
Linear Optimization Problem ◽  
Optimal Value ◽  
Nonconvex And Nonsmooth Optimization ◽  
Bilevel Linear Optimization ◽  
Nonsmooth Optimization Problem

In this paper, we investigate the pessimistic bilevel linear optimization problem (PBLOP). Based on the lower level optimal value function and duality, the PBLOP can be transformed to a single-level while nonconvex and nonsmooth optimization problem. By use of linear optimization duality, we obtain a tractable and equivalent transformation and propose algorithms for computing global or local optimal solutions. One small example is presented to illustrate the feasibility of the method.  

Ant System Initial Parameters Distribution

1970 ◽  
Vol 110 (4) ◽  
pp. 85-88
Author(s):  
R. Laptik
Keyword(s):  
Uniform Distribution ◽  
Optimal Parameter ◽  
Convergence Speed ◽  
Initial Parameter ◽  
Distributed Parameter ◽  
Optimal Solutions ◽  
Ant System ◽  
Mean Error ◽  
Optimal Value ◽  
Parameter Values

The paper presents some preliminary results on distributing initial parameter values among ants in Ant System. Two cases are studied, one with uniform distribution of heuristic information sensitivity values among ants and other with Gaussian distribution. Experimental analysis is performed by comparing behavior of Ant System with possible parameter values from the range to distributed parameter values solving Traveling Salesman Problem. Minimum mean error found, number of optimal solutions found and convergence speed are used as main indicators of Ant System performance evaluation. Uniform distribution proved to have minimum mean error difference from optimal value not more than 2 % with similar convergence speed. Number of optimal solutions found is from 2 to 5 times worse than in case of optimal parameter value due to small number of ants with parameter set close to optimal. Distribution of initial parameter values proved to be competitive when system complexity increase is not possible and initial values are not known. Bibl. 12, tabl. 2 (in English; abstracts in English and Lithuanian).http://dx.doi.org/10.5755/j01.eee.110.4.294

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