scholarly journals Extending the Scope of Robust Quadratic Optimization

2021 ◽  
Author(s):  
Ahmadreza Marandi ◽  
Aharon Ben-Tal ◽  
Dick den Hertog ◽  
Bertrand Melenberg
Keyword(s):  
Optimization Problem ◽  
Quadratic Optimization ◽  
Uncertain Parameters ◽  
Quadratic Constraints ◽  
Sample Mean ◽  
Quadratic Optimization Problem ◽  
Statistical Confidence ◽  
Robust Counterpart ◽  
Uncertainty Set ◽  
Portfolio Optimization Problem

We derive computationally tractable formulations of the robust counterparts of convex quadratic and conic quadratic constraints that are concave in matrix-valued uncertain parameters. We do this for a broad range of uncertainty sets. Our results provide extensions to known results from the literature. We also consider hard quadratic constraints: those that are convex in uncertain matrix-valued parameters. For the robust counterpart of such constraints, we derive inner and outer tractable approximations. As an application, we show how to construct a natural uncertainty set based on a statistical confidence set around a sample mean vector and covariance matrix and use this to provide a tractable reformulation of the robust counterpart of an uncertain portfolio optimization problem. We also apply the results of this paper to norm approximation problems. Summary of Contribution: This paper develops new theoretical results and algorithms that extend the scope of a robust quadratic optimization problem. More specifically, we derive computationally tractable formulations of the robust counterparts of convex quadratic and conic quadratic constraints that are concave in matrix-valued uncertain parameters. We also consider hard quadratic constraints: those that are convex in uncertain matrix-valued parameters. For the robust counterpart of such constraints, we derive inner and outer tractable approximations.

scholarly journals SDO and LDO relaxation approaches to complex fractional quadratic optimization

2020 ◽  
Author(s):  
ali ashrafi ◽  
Arezu Zare
Keyword(s):  
Optimization Problem ◽  
Original Problem ◽  
Quadratic Optimization ◽  
Parametric Quadratic Programming ◽  
Quadratic Constraints ◽  
Parametric Problem ◽  
Quadratic Optimization Problem ◽  
Lagrangian Dual ◽  
Fractional Quadratic Optimization ◽  
Generalized Newton

This paper examines a complex fractional quadratic optimization problem subject to two quadratic constraints. The original problem is transformed into a parametric quadratic programming problem by the well-known classical Dinkelbach method. Then a semidefinite and Lagrangian dual optimization approaches are presented to solve the nonconvex parametric problem at each iteration of the bisection and generalized Newton algorithms. Finally, the numerical results demonstrate the effectiveness of the proposed approaches.

Trust Your Data or Not—StQP Remains StQP: Community Detection via Robust Standard Quadratic Optimization

2020 ◽  
Author(s):  
Immanuel M. Bomze ◽  
Michael Kahr ◽  
Markus Leitner
Keyword(s):  
Community Detection ◽  
Optimization Problem ◽  
Quadratic Optimization ◽  
Quadratic Optimization Problem ◽  
Standard Quadratic Optimization ◽  
Uncertainty Sets ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Order Intervals ◽  
Copositive Matrix

We consider the robust standard quadratic optimization problem (RStQP), in which an uncertain (possibly indefinite) quadratic form is optimized over the standard simplex. Following most approaches, we model the uncertainty sets by balls, polyhedra, or spectrahedra, more generally, by ellipsoids or order intervals intersected with subcones of the copositive matrix cone. We show that the copositive relaxation gap of the RStQP equals the minimax gap under some mild assumptions on the curvature of the aforementioned uncertainty sets and present conditions under which the RStQP reduces to the standard quadratic optimization problem. These conditions also ensure that the copositive relaxation of an RStQP is exact. The theoretical findings are accompanied by the results of computational experiments for a specific application from the domain of graph clustering, more precisely, community detection in (social) networks. The results indicate that the cardinality of communities tend to increase for ellipsoidal uncertainty sets and to decrease for spectrahedral uncertainty sets.

scholarly journals Virtual Strategy Engineer: Using Artificial Neural Networks for Making Race Strategy Decisions in Circuit Motorsport

2020 ◽  
Vol 10 (21) ◽  
pp. 7805
Author(s):  
Alexander Heilmeier ◽  
André Thomaser ◽  
Michael Graf ◽  
Johannes Betz
Keyword(s):  
Neural Networks ◽  
Artificial Neural Networks ◽  
Optimization Problem ◽  
Quadratic Optimization ◽  
High Class ◽  
Simplified Approach ◽  
Quadratic Optimization Problem ◽  
Timing Data ◽  
Strategic Options ◽  
Artificial Neural

In circuit motorsport, race strategy helps to finish the race in the best possible position by optimally determining the pit stops. Depending on the racing series, pit stops are needed to replace worn-out tires, refuel the car, change drivers, or repair the car. Assuming a race without opponents and considering only tire degradation, the optimal race strategy can be determined by solving a quadratic optimization problem, as shown in the paper. In high-class motorsport, however, this simplified approach is not sufficient. There, comprehensive race simulations are used to evaluate the outcome of different strategic options. The published race simulations require the user to specify the expected strategies of all race participants manually. In such simulations, it is therefore desirable to automate the strategy decisions, for better handling and greater realism. It is against this background that we present a virtual strategy engineer (VSE) based on two artificial neural networks. Since our research is focused on the Formula 1 racing series, the VSE decides whether a driver should make a pit stop and which tire compound to fit. Its training is based on timing data of the six seasons from 2014 to 2019. The results show that the VSE makes reasonable decisions and reacts to the particular race situation. The integration of the VSE into a race simulation is presented, and the effects are analyzed in an example race.

On Sufficient Global Optimality Conditions for Bivalent Quadratic Programs with Quadratic Constraints

2015 ◽  
Vol 32 (04) ◽  
pp. 1550025
Author(s):  
Yu-Jun Gong ◽  
Yong Xia
Keyword(s):  
Optimization Problem ◽  
Sufficient Conditions ◽  
Global Optimality ◽  
Dual Model ◽  
Zero Duality Gap ◽  
Quadratic Constraints ◽  
Quadratic Optimization Problem ◽  
Lagrangian Dual ◽  
Dual Bound ◽  
Global Optimality Condition

We show the recent sufficient global optimality condition for the quadratic constrained bivalent quadratic optimization problem is equivalent to verify the zero duality gap. Then, based on the optimal parametric Lagrangian dual model, we establish improved sufficient conditions by strengthening the dual bound.

scholarly journals A duality-type method for the obstacle problem

2013 ◽  
Vol 21 (3) ◽  
pp. 181-196 ◽  
Author(s):  
Diana Rodica Merlusca
Keyword(s):  
Sobolev Spaces ◽  
Obstacle Problem ◽  
Optimization Problem ◽  
Quadratic Optimization ◽  
Higher Order ◽  
Type Method ◽  
Quadratic Optimization Problem ◽  
Approximate Problem ◽  
Duality Property ◽  
New Type

Abstract Based on a duality property, we solve the obstacle problem on Sobolev spaces of higher order. We have considered a new type of approximate problem and with the help of the duality we reduce it to a quadratic optimization problem, which can be solved much easier.

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