scholarly journals Hybrid strategies using linear and piecewise-linear decision rules for multistage adaptive linear optimization

2021 ◽  
Vol 290 (3) ◽  
pp. 1014-1030
Author(s):  
Said Rahal ◽  
Dimitri J. Papageorgiou ◽  
Zukui Li
Keyword(s):  
Piecewise Linear ◽  
Linear Optimization ◽  
Decision Rules ◽  
Linear Decision Rules

Robust Optimization for Models with Uncertain Second-Order Cone and Semidefinite Programming Constraints

2021 ◽  
Author(s):  
Jianzhe Zhen ◽  
Frans J. C. T. de Ruiter ◽  
Ernst Roos ◽  
Dick den Hertog
Keyword(s):  
Robust Optimization ◽  
Semidefinite Programming ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Decision Rules ◽  
Second Order ◽  
Minimum Volume ◽  
Second Order Cone ◽  
Linear Decision Rules ◽  
Effectiveness And Efficiency

In this paper, we consider uncertain second-order cone (SOC) and semidefinite programming (SDP) constraints with polyhedral uncertainty, which are in general computationally intractable. We propose to reformulate an uncertain SOC or SDP constraint as a set of adjustable robust linear optimization constraints with an ellipsoidal or semidefinite representable uncertainty set, respectively. The resulting adjustable problem can then (approximately) be solved by using adjustable robust linear optimization techniques. For example, we show that if linear decision rules are used, then the final robust counterpart consists of SOC or SDP constraints, respectively, which have the same computational complexity as the nominal version of the original constraints. We propose an efficient method to obtain good lower bounds. Moreover, we extend our approach to other classes of robust optimization problems, such as nonlinear problems that contain wait-and-see variables, linear problems that contain bilinear uncertainty, and general conic constraints. Numerically, we apply our approach to reformulate the problem on finding the minimum volume circumscribing ellipsoid of a polytope and solve the resulting reformulation with linear and quadratic decision rules as well as Fourier-Motzkin elimination. We demonstrate the effectiveness and efficiency of the proposed approach by comparing it with the state-of-the-art copositive approach. Moreover, we apply the proposed approach to a robust regression problem and a robust sensor network problem and use linear decision rules to solve the resulting adjustable robust linear optimization problems, which solve the problem to (near) optimality. Summary of Contribution: Computing robust solutions for nonlinear optimization problems with uncertain second-order cone and semidefinite programming constraints are of much interest in real-life applications, yet they are in general computationally intractable. This paper proposes a computationally tractable approximation for such problems. Extensive computational experiments on (i) computing the minimum volume circumscribing ellipsoid of a polytope, (ii) robust regressions, and (iii) robust sensor networks are conducted to demonstrate the effectiveness and efficiency of the proposed approach.

scholarly journals A Novel Approach for Solving Nonsmooth Optimization Problems with Application to Nonsmooth Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Hamid Reza Erfanian ◽  
M. H. Noori Skandari ◽  
A. V. Kamyad
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Piecewise Linear Function ◽  
Nonsmooth Equations ◽  
Smooth Functions ◽  
Generalized Derivative ◽  
Novel Approach

We present a new approach for solving nonsmooth optimization problems and a system of nonsmooth equations which is based on generalized derivative. For this purpose, we introduce the first order of generalized Taylor expansion of nonsmooth functions and replace it with smooth functions. In other words, nonsmooth function is approximated by a piecewise linear function based on generalized derivative. In the next step, we solve smooth linear optimization problem whose optimal solution is an approximate solution of main problem. Then, we apply the results for solving system of nonsmooth equations. Finally, for efficiency of our approach some numerical examples have been presented.

Feasibility of linear decision rules for hydropower scheduling

2014 ◽  
Author(s):  
Ida Gronvik ◽  
Ajla Hadziomerovic ◽  
Nina Ingvoldstad ◽  
Ruud Egging ◽  
Stein-Erik Fleten
Keyword(s):  
Decision Rules ◽  
Linear Decision Rules ◽  
Hydropower Scheduling

scholarly journals Linear Decision Rules for Seasonal Hydropower Planning: Modelling Considerations

2016 ◽  
Vol 87 ◽  
pp. 28-35 ◽  
Author(s):  
Simen V. Braaten ◽  
Ola Gjønnes ◽  
Knut Hjertvik ◽  
Stein-Erik Fleten
Keyword(s):  
Decision Rules ◽  
Hydropower Planning ◽  
Linear Decision Rules

A Note on Linear Decision Rules

1970 ◽  
Vol 6 (6) ◽  
pp. 1789-1790 ◽  
Author(s):  
Satish C. Nayak ◽  
Sant R. Arora
Keyword(s):  
Decision Rules ◽  
Linear Decision Rules
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