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First, strict inequalities make a problem. To illustrate it, let us consider

Minimize[{x, 1 < x && x < 2}, x]

Minimize::wksol: Warning: there is no minimum in the region in which the objective function is defined and the constraints are satisfied; returning a result on the boundary.
{1,{x->1}}

Second, the problem is too complicated for current CASes. To demostrate it, let us consider a simplified problem with weak inequalities instead of strict ones:

{(4 d k (1 - 2 r) r + k^2 r (-1 + 3 r) +  d^2 (1 + r (-5 + 6 r)))/(2 (-1 + r)^2 r) +  
q*(d^2 (2 - 3 r)+d k (-1 + 2 r) + k^2 (1+(-3 + r) r))/(-1 + r)^2, (2 r - 1)/r d <= k <=  d}/.
{d -> 1/2, q -> 3/2};
Minimize[%, r]

which produces a piecewise expression in k including

$$ \text{Root}\left[2048 \text{$\#$1}^3+\text{$\#$1}^2 \left(-7168 k^2+2048 k-3328\right)+\text{$\#$1} \left(8192 k^4-5120 k^3+9344 k^2-2304 k+1504\right)-3072 k^6+3072 k^5-6784 k^4+2880 k^3-2548 k^2+516 k-117\&,2\right]$$

Addition. The following

Minimize[{(4 d k (1 - 2 r) r + k^2 r (-1 + 3 r) + 
   d^2 (1 + r (-5 + 6 r)))/(2 (-1 + r)^2 r) + 
q*(d^2 (2 - 3 r) + 2 d k (-1 + 2 r) + 
    k^2 (1 + (-3 + r) r))/(-1 + r)^2, (2 r - 1)/r d < k < d && 
q > 1} /. {d -> 1/2}, r]

works. The result is a piecewise expression which is too long to be cited here.

First, strict inequalities make a problem. To illustrate it, let us consider

Minimize[{x, 1 < x && x < 2}, x]

Minimize::wksol: Warning: there is no minimum in the region in which the objective function is defined and the constraints are satisfied; returning a result on the boundary.
{1,{x->1}}

Second, the problem is too complicated for current CASes. To demostrate it, let us consider a simplified problem with weak inequalities instead of strict ones:

{(4 d k (1 - 2 r) r + k^2 r (-1 + 3 r) +  d^2 (1 + r (-5 + 6 r)))/(2 (-1 + r)^2 r) +  
q*(d^2 (2 - 3 r)+d k (-1 + 2 r) + k^2 (1+(-3 + r) r))/(-1 + r)^2, (2 r - 1)/r d <= k <=  d}/.
{d -> 1/2, q -> 3/2};
Minimize[%, r]

which produces a piecewise expression in k including

$$ \text{Root}\left[2048 \text{$\#$1}^3+\text{$\#$1}^2 \left(-7168 k^2+2048 k-3328\right)+\text{$\#$1} \left(8192 k^4-5120 k^3+9344 k^2-2304 k+1504\right)-3072 k^6+3072 k^5-6784 k^4+2880 k^3-2548 k^2+516 k-117\&,2\right]$$

Addition. The following

ClearAll[d, k, q, r]; {(4 d k (1 - 2 r) r + k^2 r (-1 + 3 r) + 
  d^2 (1 + r (-5 + 6 r)))/(2 (-1 + r)^2 r) + 
 q*(d^2 (2 - 3 r) + d k (-1 + 2 r) + k^2 (1 + (-3 + r) r))/(-1 + r)^2, (2 r - 1)/r d <= k <= d && 
 r > 1/2 && r < 1} /. {d -> 1/2, q -> 3/2};Minimize[%, r]

works. The result is a piecewise expression which is too long to be cited here.

Edit. Missed inequalities r > 1/2 && r < 1 are added.

First, strict inequalities make a problem. To illustrate it, let us consider

Minimize[{x, 1 < x && x < 2}, x]

Minimize::wksol: Warning: there is no minimum in the region in which the objective function is defined and the constraints are satisfied; returning a result on the boundary.
{1,{x->1}}

Second, the problem is too complicated for current CASes. To demostrate it, let us consider a simplified problem with weak inequalities instead of strict ones:

{(4 d k (1 - 2 r) r + k^2 r (-1 + 3 r) +  d^2 (1 + r (-5 + 6 r)))/(2 (-1 + r)^2 r) +  
q*(d^2 (2 - 3 r)+d k (-1 + 2 r) + k^2 (1+(-3 + r) r))/(-1 + r)^2, (2 r - 1)/r d <= k <=  d}/.
{d -> 1/2, q -> 3/2};
Minimize[%, r]

which produces a piecewise expression in k including

$$ \text{Root}\left[2048 \text{$\#$1}^3+\text{$\#$1}^2 \left(-7168 k^2+2048 k-3328\right)+\text{$\#$1} \left(8192 k^4-5120 k^3+9344 k^2-2304 k+1504\right)-3072 k^6+3072 k^5-6784 k^4+2880 k^3-2548 k^2+516 k-117\&,2\right]$$

First, strict inequalities make a problem. To illustrate it, let us consider

Minimize[{x, 1 < x && x < 2}, x]

Minimize::wksol: Warning: there is no minimum in the region in which the objective function is defined and the constraints are satisfied; returning a result on the boundary.
{1,{x->1}}

Second, the problem is too complicated for current CASes. To demostrate it, let us consider a simplified problem with weak inequalities instead of strict ones:

{(4 d k (1 - 2 r) r + k^2 r (-1 + 3 r) +  d^2 (1 + r (-5 + 6 r)))/(2 (-1 + r)^2 r) +  
q*(d^2 (2 - 3 r)+d k (-1 + 2 r) + k^2 (1+(-3 + r) r))/(-1 + r)^2, (2 r - 1)/r d <= k <=  d}/.
{d -> 1/2, q -> 3/2};
Minimize[%, r]

which produces a piecewise expression in k including

$$ \text{Root}\left[2048 \text{$\#$1}^3+\text{$\#$1}^2 \left(-7168 k^2+2048 k-3328\right)+\text{$\#$1} \left(8192 k^4-5120 k^3+9344 k^2-2304 k+1504\right)-3072 k^6+3072 k^5-6784 k^4+2880 k^3-2548 k^2+516 k-117\&,2\right]$$

Addition. The following

Minimize[{(4 d k (1 - 2 r) r + k^2 r (-1 + 3 r) + 
   d^2 (1 + r (-5 + 6 r)))/(2 (-1 + r)^2 r) + 
q*(d^2 (2 - 3 r) + 2 d k (-1 + 2 r) + 
    k^2 (1 + (-3 + r) r))/(-1 + r)^2, (2 r - 1)/r d < k < d && 
q > 1} /. {d -> 1/2}, r]

works. The result is a piecewise expression which is too long to be cited here.

First, strict inequalities make a problem. To illustrate it, let us consider

Minimize[{x, 1 < x && x < 2}, x]

Minimize::wksol: Warning: there is no minimum in the region in which the objective function is defined and the constraints are satisfied; returning a result on the boundary.
{1,{x->1}}

Second, the problem is too complicated for current CASes. To demostrate it. let us consider a simplified problem:

{(4 d k (1 - 2 r) r + k^2 r (-1 + 3 r) +  d^2 (1 + r (-5 + 6 r)))/(2 (-1 + r)^2 r) +  
q*(d^2 (2 - 3 r)+d k (-1 + 2 r) + k^2 (1+(-3 + r) r))/(-1 + r)^2, (2 r - 1)/r d <= k <=  d}/.
{d -> 1/2, q -> 3/2};
Minimize[%, r]

which produces a piecewise expression in k with

$$ \text{Root}\left[2048 \text{$\#$1}^3+\text{$\#$1}^2 \left(-7168 k^2+2048 k-3328\right)+\text{$\#$1} \left(8192 k^4-5120 k^3+9344 k^2-2304 k+1504\right)-3072 k^6+3072 k^5-6784 k^4+2880 k^3-2548 k^2+516 k-117\&,2\right]$$

First, strict inequalities make a problem. To illustrate it, let us consider

Minimize[{x, 1 < x && x < 2}, x]

Minimize::wksol: Warning: there is no minimum in the region in which the objective function is defined and the constraints are satisfied; returning a result on the boundary.
{1,{x->1}}

Second, the problem is too complicated for current CASes. To demostrate it, let us consider a simplified problem with weak inequalities instead of strict ones:

{(4 d k (1 - 2 r) r + k^2 r (-1 + 3 r) +  d^2 (1 + r (-5 + 6 r)))/(2 (-1 + r)^2 r) +  
q*(d^2 (2 - 3 r)+d k (-1 + 2 r) + k^2 (1+(-3 + r) r))/(-1 + r)^2, (2 r - 1)/r d <= k <=  d}/.
{d -> 1/2, q -> 3/2};
Minimize[%, r]

which produces a piecewise expression in k including

$$ \text{Root}\left[2048 \text{$\#$1}^3+\text{$\#$1}^2 \left(-7168 k^2+2048 k-3328\right)+\text{$\#$1} \left(8192 k^4-5120 k^3+9344 k^2-2304 k+1504\right)-3072 k^6+3072 k^5-6784 k^4+2880 k^3-2548 k^2+516 k-117\&,2\right]$$