In this paper we survey the primary research, both theoretical and applied, in the area of robust optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multistage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.

/pdf/theory-and-applications-of-robust-optimization-3tpe0fjexc.pdf

Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.

A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. Recently developed solution methods can solve even large-scale GPs extremely efficiently and reliably; at the same time a number of practical problems, particularly in circuit design, have been found to be equivalent to (or well approximated by) GPs. Putting these two together, we get effective solutions for the practical problems. The basic approach in GP modeling is to attempt to express a practical problem, such as an engineering analysis or design problem, in GP format. In the best case, this formulation is exact; when this is not possible, we settle for an approximate formulation. This tutorial paper collects together in one place the basic background material needed to do GP modeling. We start with the basic definitions and facts, and some methods used to transform problems into GP format. We show how to recognize functions and problems compatible with GP, and how to approximate functions or data in a form compatible with GP (when this is possible). We give some simple and representative examples, and also describe some common extensions of GP, along with methods for solving (or approximately solving) them.

/pdf/a-tutorial-on-geometric-programming-47sdznbzxw.pdf

A tutorial on geometric programming

http://www.optimization-online.org/DB_FILE/2012/07/3537.pdf

Recent advances in robust optimization: An overview☆

The optimal solution of a geometric program (GP) can be sensitive to variations in the problem data. Robust geometric programming can systematically alleviate the sensitivity problem by explicitly incorporating a model of data uncertainty in a GP and optimizing for the worst-case scenario under this model. However, it is not known whether a general robust GP can be reformulated as a tractable optimization problem that interior-point or other algorithms can efficiently solve. In this paper we propose an approximation method that seeks a compromise between solution accuracy and computational efficiency.

/pdf/tractable-approximate-robust-geometric-programming-49inmydp9w.pdf

Tractable approximate robust geometric programming

As the design-manufacturing interface becomes increasingly complicated with IC technology scaling, the corresponding process variability poses great challenges for nanoscale analog/RF design. Design optimization based on the enumeration of process corners has been widely used , but can suffer from inefficiency and overdesign. In this paper we propose to formulate the analog and RF design with variability problem as a special type of robust optimization problem, namely robust geometric programming. The statistical variations in both the process parameters and design variables are captured by a pre-specified confidence ellipsoid. Using such optimization with ellipsoidal uncertainy approach, robust design can be obtained with guaranteed yield bound and lower design cost, and most importantly, the problem size grows linearly with number of uncertain parameters. Numerical examples demonstrate the efficiency and reveal the trade-off between the design cost versus the yield requirement. We will also demonstrate significant improvement in the design cost using this approach compared with corner-enumeration optimization.

OPERA: optimization with ellipsoidal uncertainty for robust analog IC design

Long design cycles due to the inability to predict silicon realities are a well-known problem that plagues analog/RF integrated circuit product development. As this problem worsens for nanoscale IC technologies, the high cost of design and multiple manufacturing spins causes fewer products to have the volume required to support full-custom implementation. Design reuse and analog synthesis make analog/RF design more affordable; however, the increasing process variability and lack of modeling accuracy remain extremely challenging for nanoscale analog/RF design. We propose a regular analog/RF IC using metal-mask configurability design methodology Optimization with Recourse of Analog Circuits including Layout Extraction (ORACLE), which is a combination of reuse and shared-use by formulating the synthesis problem as an optimization with recourse problem. Using a two-stage geometric programming with recourse approach, ORACLE solves for both the globally optimal shared and application-specific variables. Furthermore, robust optimization is proposed to treat the design with variability problem, further enhancing the ORACLE methodology by providing yield bound for each configuration of regular designs. The statistical variations of the process parameters are captured by a confidence ellipsoid. We demonstrate ORACLE for regular Low Noise Amplifier designs using metal-mask configurability, where a range of applications share common underlying structure and application-specific customization is performed using the metal-mask layers. Two RF oscillator design examples are shown to achieve robust designs with guaranteed yield bound.

/pdf/regular-analog-rf-integrated-circuits-design-using-1fcr9hdl3s.pdf

Regular Analog/RF Integrated Circuits Design Using Optimization With Recourse Including Ellipsoidal Uncertainty

This paper concerns power control in lognormal fading wireless channels with correlated interference. We consider an outage-based quality of service (QoS) specification, which requires the uptime probability, i.e., the probability that no transmitter/receiver pair experiences any outage due to fading, is kept above a given level. The problem of finding an optimal power allocation to achieve this QoS goal over lognormal fading wireless channels can be posed as a stochastic geometric program (GP) with joint probabilistic constraints. This stochastic geometric program is extremely hard to solve in general, compared with the stochastic GP associated with Rayleigh fading channels. In this paper, we describe a suboptimal approach based on recently proposed robust geometric programming. With a good compromise between computational efficiency and accuracy, this robust GP relaxation approach finds a power allocation which meets the QoS requirement on the uptime probability. A numerical example is given to demonstrate the method.

/pdf/power-control-in-lognormal-fading-wireless-channels-with-59f6q0iymf.pdf

Power control in lognormal fading wireless channels with uptime probability specifications via robust geometric programming

In this paper, we consider a specific type of stochastic posynomial programming, which minimizes the expectation of a stochastically perturbed posynomial subject to inequalities of expectations of stochastically perturbed posynomials. Our goal is to build a computationally tractable approximation of this (typically intractable) optimization problem. We show that, under the assumption that the random perturbations are normal and narrow, these expectation-based stochastic posynomial programs can be approximated as conventional (deterministic) posynomial programs, that interior-point methods can solve efficiently.

Kan-Lin Hsiung

Papers

Tractable approximate robust geometric programming

OPERA: optimization with ellipsoidal uncertainty for robust analog IC design

Regular Analog/RF Integrated Circuits Design Using Optimization With Recourse Including Ellipsoidal Uncertainty

Power control in lognormal fading wireless channels with uptime probability specifications via robust geometric programming

A tractable approximation of expectation-based stochastic posynomial programs