Re: Re: Re: System of NonLinear Inequalities

*To*: mathgroup at smc.vnet.net*Subject*: [mg48619] Re: [mg48597] Re: [mg48564] Re: [mg48511] System of NonLinear Inequalities*From*: "Janos D. Pinter" <jdpinter at hfx.eastlink.ca>*Date*: Tue, 8 Jun 2004 00:48:10 -0400 (EDT)*References*: <200406040849.EAA23604@smc.vnet.net> <200406051119.HAA11747@smc.vnet.net> <6.0.1.1.1.20040605133437.01f906b8@pop.hfx.eastlink.ca> <200406070933.FAA10877@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Of course, I agree with Andrzej's comments. In certain cases, one needs exact solutions (which can be obtained only under special conditions). In many other cases, one needs to rely on numerical techniques: within the latter category, global scope search/optimization tools will be very useful. Janos Pinter At 06:33 AM 6/7/2004, you wrote: >Indeed, but one should note that a numerical solution both of an >equation and a non-strict inequality is a rather different thing from >an exact solution. Of course for many purposes it may be just as good >or better but for some problems the existence of a strict and not just >an approximate solution is crucial. For example, when using Morse >theory (the subject of my talk at IMS 2001) to study the topology of a >surface the existence of an 'approximate' critical point is >insufficient. The same is true of a few other areas of mathemtics. >Mathematica has powerful algebraic tools for strict solutions of both >equations and inequalities (the main being Collins's >CylindricalAlgebraicDecomposition) but naturally they are theoretically >limited to algebraic equations and practically to algebraic equations >with a small number of variables. > >Andrzej Kozlowski > > > >On 6 Jun 2004, at 01:45, Janos D. Pinter wrote: > > > > > Colleagues, > > > > systems of nonlinear equations and inequalities - under general > > analytical conditions - can be transformed into global optimization > > problems that can be solved numerically. (See e.g. my book 'Global > > Optimization in Action', Ch. 4.1.) Of course, such systems may have no > > solution, infinitely many solns, and 'anything in between'. > > > > If you are interested in a single particular soln, then try to express > > the quality of that soln by an 'objective function' and then you can > > solve a std math programming (optimization) problem. For example, you > > may search for the soln with minimal least squares error in an > > inconsistent system, or you may like to find the soln that is closest > > to the origin, etc. > > > > (The MathOptimizer Professional User Guide includes an example with > > multiple solutions to a system of nonlinear equations, and how to > > handle them numerically.) > > > > Regards, > > Janos Pinter > > _________________________________________________ > > > > Janos D. Pinter, PhD, DSc > > President & Research Scientist, PCS Inc. > > Adjunct Professor, Dalhousie University > > 129 Glenforest Drive, Halifax, NS, Canada B3M 1J2 > > Telephone: +1-(902)-443-5910 > > Fax: +1-(902)-431-5100; +1-(902)-443-5910 > > E-mail: jdpinter at hfx.eastlink.ca > > Web: www.dal.ca/~jdpinter > > www.pinterconsulting.com > > > > > > > > > > > > > > > > > > At 08:19 AM 6/5/2004, Andrzej Kozlowski wrote: > >> First of all, your inequalities are not written using Mathematica > >> syntax (you can't use square brackets in this way). But looking at > >> them > >> I see it does not matter whether you use proper syntax or not: no > >> computer program will ever solve a system of inequalites involving > >> someting like x^(2/(2 - x)). Your only chance is a human brain and > >> some > >> fantastic stroke of luck. > >> Sorry for being so unhelpful. > >> > >> Andrzej > >> > >> On 4 Jun 2004, at 17:49, maurizio lisciandra wrote: > >> > >> > Dear Friends, > >> > > >> > I tried to solve the following system of nonlinear inequalities with > >> > Mathematica 5.0: > >> > > >> > F < (1/2)*[a^(x/(2 - x))]*[x^(2/(2 - x))] && > >> > (2*a*B)^(x/2) + F - B > (2*a*F)^(x/2) && > >> > F > B > 0 && > >> > 0 <= a <= 1 && > >> > 0 < x < 2. > >> > > >> > I tried with Reduce, SolveInequality, SemiAlgebraicComponent, > >> > FindInstance, > >> > but all these function do not solve it. I tried to substitute x for > >> > some > >> > fixed value, but again I cannot solve it. The only value that I can > >> > substitute for x is 1, and in this case the solution is an empty > >> > space. I > >> > may be happy if I find a value that solves the system, although I > >> > really > >> > need for which inetervals in the variable this system is not an > >> empty > >> > space. > >> > > >> > Hope some nice Mathematica expert can help me. > >> > > >> > Cheers, > >> > > >> > Maurizio Lisciandra > >> > Trinity College > >> > Cambridge (UK) > >> > > >> > _________________________________________________________________ > >> > Ricerche online più semplici e veloci con MSN Toolbar! > >> > http://toolbar.msn.it/ > >> > > >> > > > > >

**References**:**System of NonLinear Inequalities***From:*"maurizio lisciandra" <lisciandra@hotmail.com>

**Re: System of NonLinear Inequalities***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**Re: Re: System of NonLinear Inequalities***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>