View source for Exponential transformation

Author: Daniel Garcia (ChBE 345)

Stewards: Dajun Yue and Prof. Fengqi You

Date presented: May 25, 2014

This article concerns the exponential transformation method for globally solving posynomial (or general geometric/signomial) optimization problems with nonconvex objective functions or constraints. A discussion of the method's development and use will be presented.

==Background==
Before discussing methods to solve posynomial optimization problems, a brief review of posynomials is of use. A posynomial, as defined by Duffin, Peterson, and Zener (1967) as a function of the form

<math> f(x_1,x_2,...,x_n)=\sum_{k=1}^K c_kx_1^{a_{1k}}...x_n^{a_{nk}} </math>

where the variables <math> x_i </math> and the coefficients <math> c_k </math> are positive, real numbers, and all of the exponents <math> a_{ik} </math> are real numbers. For example,

<math> f(x_1,x_2)=14x_1^{3.2}x_2^{4}+x_1^2x_2^7 </math> 

is a posynomial in two variables, and

<math> f(x_1)=1500x_1^{3/5} </math>

is a posynomial with one variable. 

It is not difficult to imagine a posynomial that is nonconvex, such as the examples above. Unfortunately, this can cause some problems when attempting to find a globally optimal solution of a posynomial program, as it is known that only convex problems ''guarantee'' a globally optimal solution.

Posynomial or geometric programming has been applied to solve problems in varied fields, such as signal circuit design, engineering design, project management, and inventory management, just to name a few. Clearly, the solution of such problems are important to the chemical engineer, and being able to globally solve such problems will equip the engineer with a powerful tool to solve a myriad of problems.

===Development===
Many researchers attempted to solve such problems starting in the 1960's and 1970's. Methods used in the day aimed to find only locally optimal solutions, and employed methods such as successive approximation of posynomials (called "condensation"), so-called "psuedo-duality" methods which use a weaker form of duality, and adapted nonlinear programming methods. While locally optimal solutions are certainly better than no solution at all, the desire to find a globally optimal solution was strong enough to spur the development of other methods for posynomial programs in the 1990's. Such methods included global optimization algorithms based on exponential variable transformations of the original posynomial program, convex relaxation of the original problem, and branch-and-bound-type methods.

The exponential transformation method was introduced by Costas Maranas and Christodoulos Floudas in 1997, and, under certain circumstances, provides a method to globally optimize certain posynomial programming problems.

==Formulation==

A posynomial program may be written as a generalized geometric problem (GGP) as so:


<math> \min G_0(t)=G_0^+(t)-G_0^-(t) </math>

subject to:

<math> G_j(t)=G_j^+(t)-G_j^-(t) \le 0, j=1,...,M </math>

<math> t_i \ge 0, i=1,...,N </math>

Where:

<math> G_j^+(t)=\sum_{k \in K_j^+} c_{jk} \prod_{i=1}^N t_i^{\alpha_{ijk}}, j=0,...,M </math> 

<math> G_j^-(t)=\sum_{k \in K_j^-} c_{jk} \prod_{i=1}^N t_i^{\alpha_{ijk}}, j=0,...,M </math>

where <math> t=(t_1,...,t_N) </math> is the positive variable vector; <math> G_j^+, G_j^-, j=0,...,M </math> are positive posynomial functions in t; <math> \alpha_{ijk} </math> are arbitrary real constant exponents; and, finally, <math> c_{jk} </math> are strictly positive coefficients. Sets <math> K_j^+, K_j^- </math> keep track of how many positively or negatively signed monomials form posynomials <math> G_j^+, G_j^- </math>, respectively. This formulation is constructed by grouping together monomials with identical signs.

One can simply apply the transformation <math> t_i=e^{z_i}, i=1,...,N </math> to obtain the transformed optimization problem:



<math> \min G_0(z)=G_0^+(z)-G_0^-(z) </math>


subject to <math> G_j(z)=G_j^+(z)=G_j^-(z) \le 0, j=1, ...,M </math>


<math> z_i^L \le z_i \le z_i^U, i=1,...,N </math>


Where:


<math> G_j^+(z)=\sum_{k \in K_j^+} c_{jk}e^ \left ( \sum_{i=1}^N \alpha_{ijk}z_i \right ) , j=0,...,M </math>


<math> G_j^-(z)=\sum_{k \in K_j^-} c_{jk}e^ \left ( \sum_{i=1}^N \alpha_{ijk}z_i \right ) , j=0,...,M </math>


Note that this reformulation results in the difference between two convex functions. Thus, the reformulation has, in theory, "convexified" the original nonconvex problem. However, since <math> z_i^L= \ln t_i^L </math>, it is necessary that the lower bound <math> t_i^L </math> be strictly positive for this reformulation to exist. Maranas and Floudas help circumvent this issue by essentially pre-scaling the original variables to ensure that their lower bounds will be positive. Thus, make sure that for each t_i:


<math> t_i^'=t_i+\max \left ( 0, -t_i^L+\epsilon \right ) , \epsilon > 0. </math> 


With this pre-scaling in place, it is evident that the reformulated problem is a convex programming problem, and a global solution can be found. A full algorithm and theoretical proofs can be found in the 1997 Maranas and Floudas paper. The reader is directed to the paper for a more rigorous and comprehensive interrogation into the theory behind the problem; this wiki article is simply meant to introduce the curious reader to the topic at large.

===Possible Applications===

Exponential transformation to convexify an objective function or constraints can be used on any geometric program that meets the criteria discussed above. As noted above, geometric programs can arise in a number of different fields and applications. 

Ron Dembo a geometric optimization problem concerning optimal reactor design. His system has eight nonconvex terms, and can be transformed with the exponential transformation method. Thus, chemical systems such as reactors, kinetics, and mass balance equations can be solved with exponential transformations: consider the following law of mass-action for any generic reaction:

<math> rate=kC_A^{0.5}C_B^{0.44} </math>

This equation is nonconvex as it stands, but makes a good candidate for exponential transformation. 


Posynomial expressions also develop in the field of electrical engineering. In 2001, Hershenson et al formulated a geometric optimization problem for the optimal construction, layout, and connections within CMOS operational amplifiers. Key trade-offs they looked to optimize included power dissipation, unity-gain bandwidth, and open-loop gain. An example of a nonconvex posynomial constraint that arose in their analysis involves the voltage constraints upon a sample transistor in the network, for example:

<math> \sqrt{\frac{I_7L_7}{\mu_pC_{ox}/2W_7} } \le V_{dd}-V_{out,max} </math> 

The square root of the variables (current, inductance) results in a nonconvex term that can be transformed with the exponential transformation. In the same vein, many other problems exist in electrical engineering that can also benefit from the exponential transformation.


Additional applications can be in the field of finance, supply chain, and project management.

==Examples==

Convexification of a few objective functions and constraints are provided in this section as illustrative examples. The resulting reformulations can be tested in GAMS using appropriate solvers to verify that there is indeed a globally optimal solution.

'''Example 1'''

Original problem: 
                  <math> \min f(t_1,t_2)=14t_1^{3.2}t_2^{4}+t_1^2t_2^7 </math>
    
             s.t. <math> t_1 \ge 0 </math>
                  <math> t_2 \ge 0 </math>


After exponential transformation, this problem becomes:

                  <math> \min f(z_1, z_2)=14e^ \left (3.2z_1+4z_2 \right ) + e^ \left ( 2z_1+7z_2 \right ) </math>

             s.t. <math> e^ \left ( z_1 \right ) \ge 0 </math>
                  <math> e^ \left ( z_2 \right ) \ge 0 </math>



'''Example 2'''

Original problem:
                  <math> \min f(t_1, t_2)=t_1^{3/5}t_2^2 </math>

             s.t. <math> \frac{t_1}{t_2} \le t_1^{0.7} </math>
          

becomes:

                  <math> \min f(z_1, z_2)=e^ \left ( 0.6z_1+2z_2 \right ) </math>

             s.t. <math> z_1-z_2 \le 0.7z_1 </math>


Note that in this case, the reformulated constraint is linear and, thus, convex.

There are a multitude of cases where this reformulation can be applied. As long as the coefficients of the original posynomial are strictly positive.


==References==
1. T.F. Edgar, D.M. Himmelblau, L.S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.

2. J. Nocedal, S.J. Wright, ''Numerical Optimization'', Springer, 2006.

3. J.F. Tsai, M.H. Lin, ''An Efficient Global Approach for Posynomial Geometric Programming Problems'', INFORMS Journal on Computing, 23 (3) pp. 483-492, 2011.

4. C.D. Maranas, C.A. Floudas, ''Global Optimization in Generalized Geometric Programming'', Computers & Chemical Engineering, 21 (4) pp. 351-369, 1997.

5. R. Dembo, ''A set of geometric programming test problems and their solutions'', Mathematical Programming, 10 (1) pp. 192-213, 1976.

6. M.D.M. Hershenson, S.P. Boyd, ''Optimal Design of a CMOS Op-Amp via Geometric Programming'', IEEE Transactions of Computer-Aided Design of Integrated Circuits and Systems, 20 (1) pp.1-21, 2001.

7. B.M. Worrall, M.A. Hall, ''The analysis of an inventory control model using psynomial geometric programming'', International Journal of Production Research 20 (5) pp. 657-667, 1982.