Geometric programming - Revision history

Wz274: /* Feasibility Analysis */

2021-12-11T19:40:17Z

Feasibility Analysis

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	==== Feasibility Analysis====		==== Feasibility Analysis====

	For the generalized geometric programming problem to be feasible, the constants need to be mutually consistent. <ref name=":0"/>		For the generalized geometric programming problem to be feasible, the constants need to be mutually consistent.<ref name=":0"/>
	where:		where:

Wz274: /* Definition */

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Definition

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	=== Definition ===		=== Definition ===
	The standard form of Geometric Programming optimization is to minimize the objective function which must be posynomial. The inequality constraints can only have the form of a posynomial less than or equal to one, and the equality constraints can only have the form of a monomial equal to one. <ref name=":0"/>		The standard form of Geometric Programming optimization is to minimize the objective function which must be posynomial. The inequality constraints can only have the form of a posynomial less than or equal to one, and the equality constraints can only have the form of a monomial equal to one.<ref name=":0"/>

	==== Standard Form ====		==== Standard Form ====

Wz274: /* Generalized Posynomial */

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Generalized Posynomial

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	==== Generalized Posynomial ====		==== Generalized Posynomial ====
	A generalized posynomial function, or a general posynomial, is defined as a function <math>f(x_i)</math> of positive variables <math>x_i </math> if it can be formed from posynomials using the operations of addition, multiplication, positive (fractional) power, and maximum <ref name=":0" />.		A generalized posynomial function, or a general posynomial, is defined as a function <math>f(x_i)</math> of positive variables <math>x_i </math> if it can be formed from posynomials using the operations of addition, multiplication, positive (fractional) power, and maximum<ref name=":0" />.

	For example:		For example:

Wz274: /* Posynomial */

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Posynomial

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	where <math>c_k</math> > 0,		where <math>c_k</math> > 0,

	This is a function of sum of monomial functions <ref name=":1" />.		This is a function of sum of monomial functions<ref name=":1" />.

	==== Generalized Posynomial ====		==== Generalized Posynomial ====

Wz274: /* Monomial function */

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Monomial function

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	where <math>c</math> > 0, <math>a_i</math> ε R		where <math>c</math> > 0, <math>a_i</math> ε R

	The exponents <math>a_i</math> of a monomial can be any real numbers, including fractional or negative, but the coefficient <math>c</math> can only be positive <ref name=":1">S. Boyd, L. Vandenberghe, ''Convex Optimization''. Cambridge University Press, 2009.</ref>.		The exponents <math>a_i</math> of a monomial can be any real numbers, including fractional or negative, but the coefficient <math>c</math> can only be positive<ref name=":1">S. Boyd, L. Vandenberghe, ''Convex Optimization''. Cambridge University Press, 2009.</ref>.

	=== Posynomial function ===		=== Posynomial function ===

Wz274: /* Introduction */

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Introduction

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	== Introduction ==		== Introduction ==

	A '''geometric programming (GP)''' is a family of non-linear optimization problems. Geometric programming optimization problems are typically not convex optimization problems. However, geometric programming optimization problems can be transformed from non-convex optimization problems to convex optimization problems given by their special properties. The convexification for geometric programming optimization problems is implemented by a mathematical transformation of the objective function and constraint functions and a change of decision variables. Though a number of practical problems are not equivalent to geometric programming, geometric programming is generally considered an effective solution for practical problems and it is used to well approximate and analyze various large-scale applications. Typical applications include but are not limited to optimizing power control in communication systems <ref name=":3">M. Chiang, C. W. Tan, D. P. Palomar, D. O’Neill, and D. Julian,[https://www.princeton.edu/~chiangm/gppower.pdf "Power Control By Geometric Programming"] (JULY 2007).</ref>, optimizing doping profile in semiconductor device engineering <ref name=":4" />, optimizing electronic component sizing in IC design <ref name=":0">S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi.[https://link.springer.com/article/10.1007/s11081-007-9001-7 "A Tutorial on Geometric Programming."] Retrieved 20 October 2019.</ref>, and optimizing aircraft design in aerospace engineering <ref>W. Hoburg and P. Abbeel. ''Geometric programming for aircraft design optimization.'' AIAA Journal 52.11 (2014): 2414-2426.</ref>.		A '''geometric programming (GP)''' is a family of non-linear optimization problems. Geometric programming optimization problems are typically not convex optimization problems. However, geometric programming optimization problems can be transformed from non-convex optimization problems to convex optimization problems given by their special properties. The convexification for geometric programming optimization problems is implemented by a mathematical transformation of the objective function and constraint functions and a change of decision variables. Though a number of practical problems are not equivalent to geometric programming, geometric programming is generally considered an effective solution for practical problems and it is used to well approximate and analyze various large-scale applications. Typical applications include but are not limited to optimizing power control in communication systems<ref name=":3">M. Chiang, C. W. Tan, D. P. Palomar, D. O’Neill, and D. Julian,[https://www.princeton.edu/~chiangm/gppower.pdf "Power Control By Geometric Programming"] (JULY 2007).</ref>, optimizing doping profile in semiconductor device engineering<ref name=":4" />, optimizing electronic component sizing in IC design<ref name=":0">S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi.[https://link.springer.com/article/10.1007/s11081-007-9001-7 "A Tutorial on Geometric Programming."] Retrieved 20 October 2019.</ref>, and optimizing aircraft design in aerospace engineering<ref>W. Hoburg and P. Abbeel. ''Geometric programming for aircraft design optimization.'' AIAA Journal 52.11 (2014): 2414-2426.</ref>.

	== Theory/Methodology ==		== Theory/Methodology ==

Wz274: /* Feasibility Analysis */

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Feasibility Analysis

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	==== Feasibility Analysis====		==== Feasibility Analysis====

	For the generalized geometric programming problem to be feasible, the constants need to be mutually consistent.<ref name=":0"/>		For the generalized geometric programming problem to be feasible, the constants need to be mutually consistent. <ref name=":0"/>
	where:		where:

Wz274: /* Definition */

2021-12-11T19:36:38Z

Definition

← Older revision		Revision as of 15:36, 11 December 2021
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	=== Definition ===		=== Definition ===
	The standard form of Geometric Programming optimization is to minimize the objective function which must be posynomial. The inequality constraints can only have the form of a posynomial less than or equal to one, and the equality constraints can only have the form of a monomial equal to one <ref name=":0"/>		The standard form of Geometric Programming optimization is to minimize the objective function which must be posynomial. The inequality constraints can only have the form of a posynomial less than or equal to one, and the equality constraints can only have the form of a monomial equal to one. <ref name=":0"/>

	==== Standard Form ====		==== Standard Form ====

Sgo23 at 19:30, 11 December 2021

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← Older revision		Revision as of 15:30, 11 December 2021
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	== Introduction ==		== Introduction ==

	A '''geometric programming (GP)''' is a family of non-linear optimization problems. Geometric programming optimization problems are typically not convex optimization problems. However, geometric programming optimization problems can be transformed from non-convex optimization problems to convex optimization problems given by their special properties. The convexification for geometric programming optimization problems is implemented by a mathematical transformation of the objective function and constraint functions and a change of decision variables. Though a number of practical problems are not equivalent to geometric programming, geometric programming is generally considered an effective solution for practical problems and it is used to well approximate and analyze various large-scale applications. Typical applications include but are not limited to optimizing power control in communication systems <ref name=":3">M. Chiang, C. W. Tan, D. P. Palomar, D. O’Neill, and D. Julian,[https://www.princeton.edu/~chiangm/gppower.pdf "Power Control By Geometric Programming"] (JULY 2007).</ref>, optimizing doping profile in semiconductor device engineering <ref name=":4" />, optimizing electronic component sizing in IC design <ref name=":0">S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi.[https://link.springer.com/article/10.1007/s11081-007-9001-7 "A Tutorial on Geometric Programming."] ~~''''~~ Retrieved 20 October 2019.</ref>, and optimizing aircraft design in aerospace engineering <ref>W. Hoburg and P. Abbeel. ''Geometric programming for aircraft design optimization.'' AIAA Journal 52.11 (2014): 2414-2426.</ref>.		A '''geometric programming (GP)''' is a family of non-linear optimization problems. Geometric programming optimization problems are typically not convex optimization problems. However, geometric programming optimization problems can be transformed from non-convex optimization problems to convex optimization problems given by their special properties. The convexification for geometric programming optimization problems is implemented by a mathematical transformation of the objective function and constraint functions and a change of decision variables. Though a number of practical problems are not equivalent to geometric programming, geometric programming is generally considered an effective solution for practical problems and it is used to well approximate and analyze various large-scale applications. Typical applications include but are not limited to optimizing power control in communication systems <ref name=":3">M. Chiang, C. W. Tan, D. P. Palomar, D. O’Neill, and D. Julian,[https://www.princeton.edu/~chiangm/gppower.pdf "Power Control By Geometric Programming"] (JULY 2007).</ref>, optimizing doping profile in semiconductor device engineering <ref name=":4" />, optimizing electronic component sizing in IC design <ref name=":0">S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi.[https://link.springer.com/article/10.1007/s11081-007-9001-7 "A Tutorial on Geometric Programming."] Retrieved 20 October 2019.</ref>, and optimizing aircraft design in aerospace engineering <ref>W. Hoburg and P. Abbeel. ''Geometric programming for aircraft design optimization.'' AIAA Journal 52.11 (2014): 2414-2426.</ref>.

	== Theory/Methodology ==		== Theory/Methodology ==

Sgo23: /* Applications */

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Applications

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	The floor planning problem is a generalized geometric programming problem instead of a geometric programming problem. However, any generalized geometric programming problem can be mechanically transferred to any geometric programming problem. The objective of the problem is to minimize the "bounding box" area that can include all the required objects, subject to the area of the individual object and the aspect ratio of the individual object.<ref name=":5">B. Nadhindla, H. Kakarlai, V. Gunjan, M .Reddy, Dr. F. Shaik, Geometric Programming-Based Automation of Floorplanning in ASIC Physical Design: Proceedings of the International Conference on Communications and Cyber Physical Engineering 2018. 10.1007/978-981-13-0212-1_48..</ref>		The floor planning problem is a generalized geometric programming problem instead of a geometric programming problem. However, any generalized geometric programming problem can be mechanically transferred to any geometric programming problem. The objective of the problem is to minimize the "bounding box" area that can include all the required objects, subject to the area of the individual object and the aspect ratio of the individual object.<ref name=":5">B. Nadhindla, H. Kakarlai, V. Gunjan, M .Reddy, Dr. F. Shaik, Geometric Programming-Based Automation of Floorplanning in ASIC Physical Design: Proceedings of the International Conference on Communications and Cyber Physical Engineering 2018. 10.1007/978-981-13-0212-1_48..</ref>

	[[File:Longest path.png\|thumb]"Digital Circuit"]		[[File:Longest path.png\|thumb\|Digital Circuit]]
	=== Digital Gate Sizing ===		=== Digital Gate Sizing ===
	The digital gate sizing problem is a generalized geometric programming problem instead of a geometric programming problem. However, any generalized geometric programming problem can be mechanically transferred to any geometric programming problem.<ref name=":7">S. -P. Boyd, S.-J. Kim, D. -D. Patil, M. -A. Horowitz,[https://web.stanford.edu/~boyd/papers/pdf/gp_digital_ckt.pdf "Digital Circuit Optimization via Geometric Programming. Operations Research"] (2005) 53(6):899-932. </ref> The objective of the problem is to minimize the worst-case delay of the digital signal traveling along any path through the circuit, subject to limits on the total area and power. The problem can be demonstrated as follows: To the right there is a Digital Circuit with the critical path highlighted in red. Every IC component is given power, area, resistance, and capacitance values. Using the 3 different component types: AND, OR, NOT, the total power, total area,		The digital gate sizing problem is a generalized geometric programming problem instead of a geometric programming problem. However, any generalized geometric programming problem can be mechanically transferred to any geometric programming problem.<ref name=":7">S. -P. Boyd, S.-J. Kim, D. -D. Patil, M. -A. Horowitz,[https://web.stanford.edu/~boyd/papers/pdf/gp_digital_ckt.pdf "Digital Circuit Optimization via Geometric Programming. Operations Research"] (2005) 53(6):899-932. </ref> The objective of the problem is to minimize the worst-case delay of the digital signal traveling along any path through the circuit, subject to limits on the total area and power. The problem can be demonstrated as follows: To the right there is a Digital Circuit with the critical path highlighted in red. Every IC component is given power, area, resistance, and capacitance values. Using the 3 different component types: AND, OR, NOT, the total power, total area,