# A probabilistic ellipsoid algorithm for linear optimization problems with uncertain LMI constraints

Armin Ataei, Qian Wang

Research output: Contribution to journalArticle

2 Citations (Scopus)

### Abstract

In this paper, a probabilistic algorithm based on the deep cut ellipsoid method is proposed to solve a linear optimization problem subject to an uncertain linear matrix inequality (LMI). First, a deep cut ellipsoid algorithm is introduced to address probabilistic feasibility of the uncertain LMI. Objective cuts are then defined to search for the optimal solution. The final probabilistic ellipsoid algorithm is a combination of feasibility cuts and objective cuts. It is shown that in a finite number of iterations, the ellipsoid algorithm either returns a suboptimal probabilistically feasible solution with a high confidence level or finds the problem infeasible. Furthermore, the bounds of the suboptimal value are provided with probabilistic guarantees.

Original language English (US) 248-254 7 Automatica 52 https://doi.org/10.1016/j.automatica.2014.11.010 Published - Feb 1 2015

### Fingerprint

Linear matrix inequalities

### All Science Journal Classification (ASJC) codes

• Control and Systems Engineering
• Electrical and Electronic Engineering

### Cite this

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abstract = "In this paper, a probabilistic algorithm based on the deep cut ellipsoid method is proposed to solve a linear optimization problem subject to an uncertain linear matrix inequality (LMI). First, a deep cut ellipsoid algorithm is introduced to address probabilistic feasibility of the uncertain LMI. Objective cuts are then defined to search for the optimal solution. The final probabilistic ellipsoid algorithm is a combination of feasibility cuts and objective cuts. It is shown that in a finite number of iterations, the ellipsoid algorithm either returns a suboptimal probabilistically feasible solution with a high confidence level or finds the problem infeasible. Furthermore, the bounds of the suboptimal value are provided with probabilistic guarantees.",
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In: Automatica, Vol. 52, 01.02.2015, p. 248-254.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A probabilistic ellipsoid algorithm for linear optimization problems with uncertain LMI constraints

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AU - Wang, Qian

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N2 - In this paper, a probabilistic algorithm based on the deep cut ellipsoid method is proposed to solve a linear optimization problem subject to an uncertain linear matrix inequality (LMI). First, a deep cut ellipsoid algorithm is introduced to address probabilistic feasibility of the uncertain LMI. Objective cuts are then defined to search for the optimal solution. The final probabilistic ellipsoid algorithm is a combination of feasibility cuts and objective cuts. It is shown that in a finite number of iterations, the ellipsoid algorithm either returns a suboptimal probabilistically feasible solution with a high confidence level or finds the problem infeasible. Furthermore, the bounds of the suboptimal value are provided with probabilistic guarantees.

AB - In this paper, a probabilistic algorithm based on the deep cut ellipsoid method is proposed to solve a linear optimization problem subject to an uncertain linear matrix inequality (LMI). First, a deep cut ellipsoid algorithm is introduced to address probabilistic feasibility of the uncertain LMI. Objective cuts are then defined to search for the optimal solution. The final probabilistic ellipsoid algorithm is a combination of feasibility cuts and objective cuts. It is shown that in a finite number of iterations, the ellipsoid algorithm either returns a suboptimal probabilistically feasible solution with a high confidence level or finds the problem infeasible. Furthermore, the bounds of the suboptimal value are provided with probabilistic guarantees.

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