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Design by Optimization
Published in Wai-Kai Chen, Computer Aided Design and Design Automation, 2018
For example, the function 3.7x11.4x23+1.8x2−1x32.3 is a posynomial in the variables x1, x2, and x3. Roughly speaking, a posynomial is a function that is similar to a polynomial, except that the coeffcients γj must be positive, and an exponent αij could be any real number, and not necessarily a positive integer, unlike a polynomial. A posynomial has the useful property that it can be mapped onto a convex function through an elementary variable transformation, (xi) = (ezi). Such functional forms are useful since in the case of an optimization problem where the objective function and the constraints are posynomial, the problem can easily be mapped onto a convex programming problem.
Wire Sizing
Published in Charles J. Alpert, Dinesh P. Mehta, Sachin S. Sapatnekar, Handbook of Algorithms for Physical Design Automation, 2008
Sanghamitra Roy, Charlie Chung-Ping Chen
The Elmore delay of an RC tree is a posynomial function of the sizes of wires in the tree. A posynomial is a function almost like a polynomial but with positive coefficients and real exponents. It can be described by the general expression t(W)=∑j=1kcj∏i=1nWiαij, where cj, j = 1… k are positive real numbers, and aij are real numbers. The transformation exi=Wi transforms any posynomial function of Wi’s to a convex function of xi’s.
Allocating outreach resources for disease control in a dynamic population with information spread
Published in IISE Transactions, 2020
Bryan Wilder, Sze-chuan Suen, Milind Tambe
We will show that is submodular, allowing us to apply principled optimization methods. Our first step is to analyze which captures the population dynamics of the SEIS model. We show that for any component i, is supermodular in We do this in two steps. First, we show that x is a posynomial in the variables A posynomial is a polynomial with entirely non-negative coefficients. Then, we will show that any function which is a posynomial in is continuous supermodular in We start by showing the following:
MOS Amplifier Design Methodology for Optimum Performance
Published in IETE Journal of Research, 2020
Abir J. Mondal, Paromita Bhattacharjee, Pinaki Chakraborty, Bidyut K. Bhattacharyya
As a matter of fact that any circuit design problem is bound to be expressed in a mathematical form, many mathematical optimization methods, such as linear programming, NLP, quadratic programming, and geometric programming, which are widely used for constrained optimization problem [11, 12] can be incorporated into any analog design methodology. Most recent works in analog circuit optimization have been done using geometric programming [13–15]. Using their synthesis method [13], Mandal et al. proposed an op-amp sizing method using geometric programming formulation [15]. Hershenson et al. also proposed a similar optimal design approach for op-amp using geometric programming [14]. Geometric programming uses circuit equations in posynomial form and it can be transformed to a convex problem by using log transformation [12]. Geometric programming ensures a global optimum solution but also limits the form of objective function since it allows only minimization of the objective function. Henceforth, an objective to be maximized has to be converted to a minimization form, which may not be flexible to be done in every case. In view of these factors, an NLP-based approach has an edge over its other counterparts. It permits the objective to take any of the minimization or maximization form, does not restrict the constraint equations, and accepts convex as well as concave problem. The methodology presented in this paper based on NLP results in a global optimal solution and is extremely fast in computation.
Finding all global optima of engineering design problems with discrete signomial terms
Published in Engineering Optimization, 2020
Jung-Fa Tsai, Ming-Hua Lin, Lei-Yi Peng
Some research has developed logarithmic methods to solve NLIP problems with signomial terms using only a logarithmic number of extra binary variables and constraints to reformulate the discrete signomial term. Li and Lu (2009) converted a generalized geometric programming problem with discrete variables into a mixed-integer linear programming (MILP) problem by convexification strategies and piecewise linearization techniques. Vielma and Nemhauser (2011) introduced mixed integer binary formulations for SOS1 and SOS2 constraints using a logarithmic number of binary variables and extra constraints as continuous variables. Tsai and Lin (2011) employed variable transformation and piecewise linearization formulation, developed by Vielma and Nemhauser (2011), to solve posynomial geometric programming problems. Lin, Tsai, and Wang (2012) applied convexification strategies and piecewise linearization techniques to solve engineering optimization problems. Tsai and Lin (2013) proposed an efficient linearization method employing a logarithmic number of extra binary variables and constraints to reformulate a signomial term with pure discrete variables. Similarly, Lu (2013) linearized the product of free-sign discrete functions into a set of linear inequalities using logarithmic numbers of binary variables and constraints. Compared with the Vielma and Nemhauser (2011) approach, Li, Huang, and Fang (2013) proposed an improved representation of binary vectors by using fewer constraints. Based on the Li, Huang, and Fang (2013) method, Lin, Tsai, and Chang (2017) solved the three-dimensional rectangular packing problems. Li et al. (2016) also proposed an enhanced logarithmic method to reformulate the signomial discrete programming problem as a mixed 0–1 linear program with fewer inequality constraints than currently known methods.