Xjyuk

Using the documentclass[a4paper, 10pt, conference]ieeeconf I have the below code that is forcing the text to move wider than the width of the column and is not being moving to next line.

beginequation
normalsize
 J_i(G)=begincases
 +infty, & textif $j$ gives the very high return and has the slope of the curve inward towards other peak points. \
 0, & textif $j$ gives the very high return and has the slope of the curve outward towards other peak points.\
 -infty, & textotherwise,
 endcases
 labeleq:eq:mylabel
endequation

This is the output I am getting.

I trtied various things, including adding \ after the of but it does not work.
Also adding linebreak in the same place does not force the move to a new line.

The second column in cases is typeset in math mode and is typically a short condition; when text is involved, text can be used, but this doesn't split copy across lines.

You can do it with a parbox. On the other hand, the result is not pretty. I suggest using a shorthand that can be explained just below the equation.

documentclass[a4paper, 10pt, conference]IEEEconf
usepackageamsmath % for 'cases' environment

usepackagelipsum % just for the example

begindocument

lipsum[1][1-2]
beginequationlabeleq:mylabel
J_i(G)=
begincases
 +infty, & parbox[t]0.5linewidth
 if $i$ gives the very high return and has the slope 
 of the curve inward towards other peak points;
 \
 0, & parbox[t]0.5linewidth
 if $i$ gives the very high return and has the slope 
 of the curve outward towards other peak points;
 \
 -infty, & textotherwise.
endcases
endequation
lipsum[1][3-4]
beginequationlabeleq:mylabel-new
J_i(G)=
begincases
 +infty, & textinward case for $i$; \
 0, & textoutward case for $i$; \
 -infty, & textother cases for $i$.
endcases
endequation
In the above equation, `inward' and `outward' case refer to when $i$ gives the
very high return and has the slope of the curve inward or, respectively, outward
towards other peak point.

lipsum[2-10]

enddocument

The usage of lipsum is only to provide context to the formulas.

As you've (re)discovered, the cases environment doesn't automatically line-wrap long explanations. As a remedy, I suggest you encase the contents of the cases environment in a custom array environment which allows line-wrapping. Note that it's not necessary to employ text directives for the explanatory textual material.

[For some reason, I can't upload a screenshot of the output that's produced by the following code. I will try again later, i.e., in a few hours.]

A general comment: I think your readers would strongly benefit from you finding a more succinct way of stating the clauses "if $j$ gives the very high return and has the slope of the curve inward towards other peak points" and "if $j$ gives the very high return and has the slope of the curve outward towards other peak points", especially since these clauses differ in only one word: "inward" in the first clause and "outward" in the second.

documentclass[a4paper, 10pt, conference]IEEEconf
usepackageamsmath % for 'cases' environment

usepackagearray % for 'newcolumntype' macro
newcolumntypeP[1]>raggedrightarraybackslashp#1
newenvironmentmyarray%
 beginarray@ l P0.5columnwidth @%
 %
 endarray

begindocument
beginequationlabeleq:eq:mylabel
J_i(G)=
begincases
 beginmyarray
 +infty, & if $j$ gives the very high return and has the slope of the curve inward towards other peak points; \
 0, & if $j$ gives the very high return and has the slope of the curve outward towards other peak points;\
 -infty, & otherwise.
 endmyarray
endcases
endequation
enddocument