Crthuky

I combined 2 columns in the last row. How do I make the text in the form of bullets, start from the beginning of the row and not leave so much blank space? Also, how do I get rid of the space at the bottom of a row? Thanks in advance!

documentclass[8pt]article
usepackagearray
usepackagepdflscape
usepackagecomment
usepackagegraphicx
usepackageeasytable
usepackageamsmath
usepackageamssymb
usepackagemathtools
usepackagerotating
usepackagemakecell
usepackagemultirow
usepackagebooktabs
usepackagemultirow,hhline,graphicx,array
usepackage[margin=0.5in]geometry

%DeclareMathSizes816168

newcommandxmathbfx
newcommandgmathbfg
newcommandhmathbfh
newcommandmathbf0 %<- that's not a good idea
newcolumntypeM[1]>centeringarraybackslashm#1

begindocument
aboverulesep=0ex
belowrulesep=0ex
%renewcommandarraystretch5
newgeometrymargin=0.1cm
beginlandscape
% Table generated by Excel2LaTeX from sheet 'Sheet1'
begintable[htbp]
 centering
 captionAdd caption
begintabularp20em
cmidrule3-5 multicolumn1c 
& 
& 
makecelltextbfUnconstrained \ $undersetxinmathbbR^n 
mathrmminimize f(x)$
&
makecelltextbfConstrained: Reduced Form \ 
$undersetxinmathbbR^nmathrmminimize f(x)$ \
$mathrmsubject to h(x)= $ 
& 
makecelltextbfConstrained: Lagrangian Form \ 
$undersetxinmathbbR^nmathrmminimize f(x)$ \
$mathrmsubject to h(x)=,g(x)leq$ 
\
midrule
multirow2*rotatebox[origin=r]90makecellLocal Optimality 
Conditions~~~~~~~~~~~~~~~~~~~~~~~~~~~~ & multicolumn1 
rotatebox[origin=r]90 First Order Necessary~~~~~~ 
& 
At a local minimizer, the gradient of the objective function must be zero 
[
nabla f(x_dagger)=
]
& 
At a local minimizer, the reduced gradient must be zero if $partial 
h/partial s$ is invertible.
[
nabla_d f_R (x_dagger)=0
]
[
h(x_dagger)=0
]
[
textwhere x= beginbmatrix
d\s
endbmatrix
,nabla_d f_R (x_dagger)=fracpartial fpartial d-fracpartial f 
partial s bigg( fracpartial hpartial s bigg )^-1fracpartial 
hpartial d
]
& 
At a local minimizer, the KKT conditions must be satisfied if the point is 
regular (i.e.: if the linear independence constraint qualification (LICQ) is 
satisfied: if $nabla h_dagger(x_*)$ has independent rows).
[
nabla _x L(x_dagger)=0 
]
[
h(x_dagger)=0,g(x_dagger)≤0
]
[
mu_dagger^⊤ g(x_dagger)=0
]
[
mu_dagger≥0
]
[
textwhere L(x_dagger)=f(x_dagger)+lambda^⊤ h(x_dagger)+μ^⊤ 
g(x_dagger) 
]
\
cmidrule2-5 multicolumn1c 
& 
multicolumn1rotatebox[origin=r]90 Second Order 
Sufficiency~~~~~~~~ 
& 
If the Hessian of the objective function is positive definite at a point 
where the gradient is zero, the point is a local minimum.
[
partial x^Tnabla^2f(x_*)partial x>0 
]
[
forall partial x neq 0
]
A Hessian matrix is positive definite if all of its eigenvalues are 
positive.
& 
If the reduced Hessian is positive definite at a point where the reduced 
gradient is zero, the point is a local minimum.
[
partial d^⊤ nabla_d^2 f_R (x_*)partial d>0, forall partial d neq 0
]
[
textwhere nabla_d^2 f_R (x_*)=A fracpartial ^2 fpartial x^2 
A^T+ fracpartial fpartial s fracpartial ^2 spartial d^2
]
[
A=
bigg[
I hspace2mmbigg(fracpartial spartial dbigg)^T
bigg]
, fracpartial^2 spartial d^2 =-bigg(fracpartial hpartial 
sbigg)^-1 A fracpartial^2 hpartial x^2 A^T
]
& 
If the Hessian of the Lagrangian is positive definite on the subspace 
tangent to the active constraints at a KKT point, the point is a local 
minimum.
[
partial x^Tnabla^2_x L(x_*)partial x>0 
 ]
 [
 forall partial x neq 0: nabla_x h_dagger(x_*)partial x = 0
 ]
 [
 textwhere h_dagger(x_*) = [h(x_*)^T, g_j(x_*)forall 
 j:mu_j>0]^T
 ]
 A Hessian matrix is positive definite on the subspace tangent to the 
 active constraints if the last n-m leading principle minors of the 
 bordered Hessian $beginbmatrix
 0 & nabla h\ nabla h^T & nabla^2_x L
 endbmatrix$have sign $(-1)^m$, where m is the number of active 
 constraints.
\
midrule
multicolumn1rotatebox[origin=r]90makecell Global Optimality Conditions~~~~~~~ 
&
multicolumn1p1.4emrotatebox[origin=r]90makecell 
Convexity~~~~~~~~~~~~~~~~~~ 
 
& 
beginitemize
item For convex functions, if a point is a local minimum it is also the 
global minimum and a local minimizer is also a global minimizer (not 
necessarily the only one).
item If the objective function is nonconvex, it may or may not have 
multiple local minima.
item A convex function* is a function whose Hessian is positive 
semidefinite for all x.
item A Hessian matrix is positive semidefinite if all of its eigenvalues 
are nonnegative.
enditemize
& 
 multicolumn1c
 &
beginitemize
item A convex optimization problem is a problem in negative null form where 
f(x) and g(x) are each convex functions and h(x) are affine functions.
item For convex optimization problems, a local minimum is also the global 
minimum, and a local minimizer is also a global minimizer (not necessarily the only one).
item A nonconvex optimization problem may or may not have multiple 
local minima and/or disconnected feasible regions.
 enditemize 
 \
 bottomrule
 endtabular%
 labeltab:addlabel%
endtable%
endlandscape
restoregeometry
enddocument

Here is an improvement: some code simplification exploiting the possibilities of makecell, enumitem and loading tabularx:

documentclass[8pt]extarticle
usepackagearray
usepackagepdflscape
usepackagecomment
usepackagegraphicx
usepackageeasytable
usepackageenumitem
usepackageamssymb
usepackagemathtools, nccmath, esdiff
usepackagerotating
usepackagemakecell
renewcommandtheadfontnormalsizebfseries
usepackagebooktabs
usepackagemultirow,hhline,graphicx,array, caption, tabularx
usepackage[margin=0.5in]geometry

newcommandxmathbfx
newcommandgmathbfg
newcommandhmathbfh
newcommandmathbf0 %<- that's not a good idea
newcolumntypeM[1]>centeringarraybackslashm#1
makeatletter
newcommand*compress@minipagetrue
makeatother
newlengthTXcolwd

begindocument
aboverulesep=0ex
belowrulesep=0ex
renewcommandtheadaligntc
newgeometrymargin=0.1cm
beginlandscape
nullvfill
% Table generated by Excel2LaTeX from sheet 'Sheet1'
begintable[htbp]
setlist[itemize, 1]wide=0pt, leftmargin=*, before=compress, after=vspace*dimexprtopsep-baselineskip
setlengthextrarowheight4pt
 centering
 captionAdd caption
begintabularxlinewidth% }% p0.7em
cmidrule3-5 multicolumn1c
& & theadUnconstrained \[1ex] $undersetx in mathbbR^n
mathrmminimize f(x)$
&
theadConstrained: Reduced Form \
$beginarraylundersetx in mathbbR^nmathrmminimize f(x) \
mathrmsubject toenspace h(x)=
endarray $
&
theadConstrained: Lagrangian Form \
$beginarraylundersetx in mathbbR^nmathrmminimize f(x) \
mathrmsubject to h(x)=,g(x)leq
endarray $ \
midrule
multirowcell20rotatebox90Local Optimality Conditions%
& 
multirowcell9rotatebox90First Order Necessary
&
 At a local minimizer, the gradient of the objective function must be zero
[ nabla f(x_dagger)= ]
& 
At a local minimizer, the reduced gradient must be zero if $partial h/partial s$ is invertible. useshortskip
begingather*
nabla_d f_R (x_dagger)=0 \
h(x_dagger)=0 \
textwhere x= beginbmatrix
d\s
endbmatrix,:nabla_d f_R (x_dagger)=fracpartial fpartial d-fracpartial f
partial s biggl( diffphs biggr )^mkern-6mu-1diffphd
endgather*
& 
At a local minimizer, the KKT conditions must be satisfied if the point is regular (i.e.: if the linear independence constraint qualification (LICQ) is satisfied: if $ nabla h_dagger(x_*)$ has independent rows).useshortskip
begingather*
nabla _x L(x_dagger)=0 \
h(x_dagger)=0,g(x_dagger) le 0 \
mu_dagger^T g(x_dagger)=0 \
mu_dagger ge 0 \
textwhere L(x_dagger)=f(x_dagger)+lambda^T h(x_dagger)+mu ^T
g(x_dagger)
endgather*
vspace*dimexpr 1ex-baselineskip \
cmidrule2-5%
& 
multirowcell11rotatebox90Second Order Sufficiency %
&
 If the Hessian of the objective function is positive definite at a point where the gradient is zero, the point is a local minimum.
 begingather*
partial x^Tnabla^2f(x_*)partial x>0 \
forall partial x neq 0
endgather*
A Hessian matrix is positive definite if all of its eigenvalues are positive.
&
 If the reduced Hessian is positive definite at a point where the reduced gradient is zero, the point is a local minimum.
begingather*
partial d^T nabla_d^2 f_R (x_*)partial d>0, forall partial d neq 0 \
textwhere nabla_d^2 f_R (x_*)=A fracpartial ^2 fpartial x^2
A^T+ diffpfs diffp[2]sd \
A= biggl[
I hspace2mmbiggl(diffpsdbiggr)^T
biggr],
fracpartial^2 spartial d^2 =-biggl(diffphsbiggr)^mkern-6mu -1 A, diffp[2]hx A^T
 endgather*
& 
If the Hessian of the Lagrangian is positive definite on the subspace tangent to the active constraints at a KKT point, the point is a local minimum.
begingather*
partial x^Tnabla^2_x L(x_*)partial x>0 \
 forall partial x neq 0: nabla_x h_dagger(x_*)partial x = 0 \
 textwhere h_dagger(x_*) = [h(x_*)^T, g_j(x_*)forall
 j:mu_j>0]^T
 endgather*
 A Hessian matrix is positive definite on the subspace tangent to the active constraints if the last $ n $-$ m $ leading principal minors of the bordered Hessian %
 $beginbmatrix
 0 & nabla h\ nabla h^T & nabla^2_x L
 endbmatrix$have sign $(-1)^m$, where $ m $ is the number of active
 constraints. smallskip
\
midrule
multirowcell9rotatebox90Global Optimality Conditions
&
multirowcell9rotatebox90Convexity
& beginitemize
item For convex functions, if a point is a local minimum it is also the global minimum and a local minimizer is also a global minimizer (not necessarily the only one).
item If the objective function is nonconvex, it may or may not have multiple local minima.
item A convex function* is a function whose Hessian is positive semidefinite for all x.
item A Hessian matrix is positive semidefinite if all of its eigenvalues are nonnegative.
enditemize
& 
multicolumn2%
beginitemize
item A convex optimization problem is a problem in negative null form where f(x) and g(x) are each convex functions and h(x) are affine functions.
item For convex optimization problems, a local minimum is also the global minimum, and a local minimizer is also a global minimizer (not necessarily the only one).
item A nonconvex optimization problem may or may not have multiple local minima and/or disconnected feasible regions.
 enditemize \
 bottomrule
 endtabularx%
 labeltab:addlabel%
endtable%
vfill
endlandscape
restoregeometry

enddocument

I think you have to use the multicolumn command differently:

multicolumn2p42em
beginitemize
item A convex optimization problem is a problem in negative null form
where f(x) and g(x) are each convex functions and h(x) are affine 
functions.
item For convex optimization problems, a local minimum is also the global
minimum, and a local minimizer is also a global minimizer (not necessarily 
the only one).
item A nonconvex optimization problem may or may not have multiple
local minima and/or disconnected feasible regions.
enditemize

See the documentation or How to merge columns in a table? when in doubt.

As to the vertical alignment, the column type m should do the trick:
p,m and b columns in tables