Mrthfx

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}

usepackage{array}

usepackage{pdflscape}

usepackage{comment}

usepackage{graphicx}

usepackage{easytable}

usepackage{amsmath}

usepackage{amssymb}

usepackage{mathtools}

usepackage{rotating}

usepackage{makecell}

usepackage{multirow}

usepackage{booktabs}

usepackage{multirow,hhline,graphicx,array}

usepackage[margin=0.5in]{geometry}



%DeclareMathSizes{8}{16}{16}{8}



newcommand{x}{mathbf{x}}

newcommand{g}{mathbf{g}}

newcommand{h}{mathbf{h}}

newcommand{}{mathbf{0}} %<- that's not a good idea

newcolumntype{M}[1]{>{centeringarraybackslash}m{#1}}



begin{document}

aboverulesep=0ex

belowrulesep=0ex

%renewcommand{arraystretch}{5}

newgeometry{margin=0.1cm}

begin{landscape}

% Table generated by Excel2LaTeX from sheet 'Sheet1'

begin{table}[htbp]

  centering

  caption{Add caption}

begin{tabular}{|p{0.7em}| p{0.7em}|p{20em}|p{21em}|p{21em}|}

cmidrule{3-5}    multicolumn{1}{c}{} 

&       

& 

makecell{textbf{Unconstrained} \ $underset{xinmathbb{R}^n}    

{mathrm{minimize}} f(x)$}

&

makecell{textbf{Constrained: Reduced Form} \                     

$underset{xinmathbb{R}^n}{mathrm{minimize}} f(x)$ \

$mathrm{subject to } h(x)= $} 

& 

makecell{textbf{Constrained: Lagrangian Form} \ 

$underset{xinmathbb{R}^n}{mathrm{minimize}} f(x)$ \

$mathrm{subject to } h(x)=,g(x)leq$ }

\

midrule

multirow{2}{*}{rotatebox[origin=r]{90}{makecell{Local Optimality 

Conditions~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}} & multicolumn{1}{p{0.7em}|}            

{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

]

[

text{where } x= begin{bmatrix}

d\s

end{bmatrix}

,nabla_d f_R (x_{dagger})=frac{partial f}{partial d}-frac{partial f}    

{partial s} bigg( frac{partial h}{partial s} bigg )^{-1}frac{partial 

h}{partial 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

]

[

text{where } L(x_{dagger})=f(x_{dagger})+lambda^⊤ h(x_{dagger})+μ^⊤     

g(x_{dagger})    

]

\

cmidrule{2-5}    multicolumn{1}{|c|}{} 

& 

multicolumn{1}{p{0.7em}|}{rotatebox[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

]

[

text{where }nabla_d^2 f_R (x_{*})=A frac{partial ^2 f}{partial x^2}     

A^{T}+ frac{partial f}{partial s} frac{partial ^2 s}{partial d^2}

]

[

A=

bigg[

I hspace{2mm}bigg({frac{partial s}{partial d}bigg)}^T

bigg]

,  frac{partial^2 s}{partial d^2} =-bigg(frac{partial h}{partial 

s}bigg)^{-1} A frac{partial^2 h}{partial 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

    ]

    [

    text{where }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 $begin{bmatrix}

 0 & nabla h\ nabla h^T & nabla^2_x L

 end{bmatrix}$have sign $(-1)^m$, where m is the number of active 

 constraints.

\

midrule

multicolumn{1}{|p{1.4em}|}{rotatebox[origin=r]{90}{makecell{ Global Optimality Conditions~~~~~~~} }} 

&

multicolumn{1}{p{1.4em}|}{rotatebox[origin=r]{90}{makecell{         

Convexity~~~~~~~~~~~~~~~~~~} 

}} 

&       

begin{itemize}

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.

end{itemize}

& 

 multicolumn{1}{c}{}

 &

begin{itemize}

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.

 end{itemize} 

 \

 bottomrule

 end{tabular}%

  label{tab:addlabel}%

end{table}%

end{landscape}

restoregeometry

end{document}

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

documentclass[8pt]{extarticle}

usepackage{array}

usepackage{pdflscape}

usepackage{comment}

usepackage{graphicx}

usepackage{easytable}

usepackage{enumitem}

usepackage{amssymb}

usepackage{mathtools, nccmath, esdiff}

usepackage{rotating}

usepackage{makecell}

renewcommand{theadfont}{normalsizebfseries}

usepackage{booktabs}

usepackage{multirow,hhline,graphicx,array, caption, tabularx}

usepackage[margin=0.5in]{geometry}



newcommand{x}{mathbf{x}}

newcommand{g}{mathbf{g}}

newcommand{h}{mathbf{h}}

newcommand{}{mathbf{0}} %<- that's not a good idea

newcolumntype{M}[1]{>{centeringarraybackslash}m{#1}}

makeatletter

newcommand*{compress}{@minipagetrue}

makeatother

newlength{TXcolwd}



begin{document}

aboverulesep=0ex

belowrulesep=0ex

renewcommand{theadalign}{tc}

newgeometry{margin=0.1cm}

begin{landscape}

nullvfill

% Table generated by Excel2LaTeX from sheet 'Sheet1'

begin{table}[htbp]

setlist[itemize, 1]{wide=0pt, leftmargin=*, before=compress, after=vspace*{dimexprtopsep-baselineskip}}

setlength{extrarowheight}{4pt}

  centering

  caption{Add caption}

begin{tabularx}{linewidth}{|c|c|X|X|X|}% }{|p{0.7em}|p{0.4em}|X|X|X|}% p{0.7em}

cmidrule{3-5} multicolumn{1}{c}{}

& & thead{Unconstrained \[1ex] $underset{x in mathbb{R}^n}

{mathrm{minimize}} f(x)$}

&

thead{Constrained: Reduced Form \

$begin{array}{l}underset{x in mathbb{R}^n}{mathrm{minimize}} f(x) \

mathrm{subject toenspace} h(x)=

end{array} $}

&

thead{Constrained: Lagrangian Form \

$begin{array}{l}underset{x in mathbb{R}^n}{mathrm{minimize}} f(x) \

mathrm{subject to } h(x)=,g(x)leq

end{array} $ } \

midrule

multirowcell{20}{rotatebox{90}{Local Optimality Conditions}}%

& 

multirowcell{9}{rotatebox{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. useshortskip

begin{gather*}

nabla_d f_R (x_{dagger})=0 \

h(x_{dagger})=0 \

text{where } x= begin{bmatrix}

d\s

end{bmatrix},:nabla_d f_R (x_{dagger})=frac{partial f}{partial d}-frac{partial f}

{partial s} biggl( diffp{h}{s} biggr )^{mkern-6mu-1}diffp{h}{d}

end{gather*}

& 

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

begin{gather*}

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 \

text{where } L(x_{dagger})=f(x_{dagger})+lambda^T h(x_{dagger})+mu ^T

g(x_{dagger})

end{gather*}

vspace*{dimexpr 1ex-baselineskip} \

cmidrule{2-5}%

& 

multirowcell{11}{rotatebox{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.

 begin{gather*}

partial x^Tnabla^2f(x_{*})partial x>0 \

forall partial x neq 0

end{gather*}

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.

begin{gather*}

partial d^T nabla_d^2 f_R (x_{*})partial d>0, forall partial d neq 0 \

text{where }nabla_d^2 f_R (x_{*})=A frac{partial ^2 f}{partial x^2}

A^{T}+ diffp{f}{s} diffp[2]{s}{d} \

A= biggl[

I hspace{2mm}biggl({diffp{s}{d}biggr)}^T

biggr],

frac{partial^2 s}{partial d^2} =-biggl(diffp{h}{s}biggr)^{mkern-6mu -1} A, diffp[2]{h}{x} A^{T}

 end{gather*}

& 

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.

begin{gather*}

partial x^Tnabla^2_x L(x_{*})partial x>0 \

 forall partial x neq 0: nabla_x h_{dagger}(x_{*})partial x = 0 \

 text{where }h_{dagger}(x_{*}) = [h(x_{*})^T, g_j(x_{*})forall

 j:mu_j>0]^T

 end{gather*}

 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 %

 $begin{bmatrix}

 0 & nabla h\ nabla h^T & nabla^2_x L

 end{bmatrix}$have sign $(-1)^m$, where $ m $ is the number of active

 constraints. smallskip

\

midrule

multirowcell{9}{rotatebox{90}{Global Optimality Conditions}}

&

multirowcell{9}{rotatebox{90}{Convexity}}

& begin{itemize}

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.

end{itemize}

& 

multicolumn{2}{p{57em}|}{%

begin{itemize}

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.

 end{itemize}} \

 bottomrule

 end{tabularx}%

  label{tab:addlabel}%

end{table}%

vfill

end{landscape}

restoregeometry



end{document}

I think you have to use the multicolumn command differently:

multicolumn{2}{p{42em}|}{

begin{itemize}

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.

end{itemize}}

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