Something to perhaps realize is that your two problems may not be as different as you think if $\lambda$ is an ad-hoc parameter.
For any solution to your 2nd problem (where $\theta > 1$), there exists a $\lambda$ for problem 1 which gives you the same solution as problem 2.
Example
Let $f$ and $g$ be convex functions and let $\mathbf{x}$ denote a vector. To build some intuition, consider the two problems (stylized versions of your problem 1 and problem 2).
Problem A:
\begin{equation}
\begin{array}{*2{>{\displaystyle}r}}
\mbox{minimize (over $\mathbf{x}$)} & f(\mathbf{x}) + \lambda_1g(\mathbf{x}) \end{array}
\end{equation}
Problem B:
Let $h(x)$ be convex and non-decreasing over the range of $g$, hence composition $h(g(\mathbf{x}))$ is also convex. Consider
\begin{equation}
\begin{array}{*2{>{\displaystyle}r}}
\mbox{minimize (over $\mathbf{x}$)} & f(\mathbf{x}) + \lambda_2 h\left( g(\mathbf{x}) \right) \end{array}
\end{equation}
First order conditions
Both are convex optimization problems and the first order conditions are necessary and sufficient.
- The first order condition for problem A is $\frac{\partial f}{\partial \mathbf{x}} + \lambda_1 \frac{\partial g}{\partial \mathbf{x}} = 0$.
- The first order condition for problem B is $\frac{\partial f}{\partial \mathbf{x}} + \lambda_2 h'(g(\mathbf{x}))\frac{\partial g}{\partial \mathbf{x}} = 0$.
Let $\mathbf{x}^*$ be the solution to either problem A or problem B. If $\lambda_1 = \lambda_2 h'(g(\mathbf{x}^*))$ then problem A and problem B have the same solution.
Your penalty is $h(x) = x^\theta$ (which is convex and increasing on the region $x >0$ if $\theta > 1$). Observe $h'(x) = \theta x^{\theta - 1}$. Also observe that to achieve the same solution $\mathbf{x}^*$, you could have $\lambda_1 > \lambda_2$ or $\lambda_1 < \lambda_2$ depending on the specific parameterization. As you've formulated the problem, increasing $\theta$ doesn't necessarily increase the penalty for larger risk (where risk is defined as the standard deviation of returns)! For example, let's say your solution $\mathbf{x}^*$ for $\theta = 1$ gives a standard deviation of returns of .01. Simply raising $\theta$ to $1000$ would increase the convexity of the risk penalty but it would decrease the level of the penalty to numerical irrelevance.
Your $g(\mathbf{x}) = \sqrt{\mathbf{x}'\Sigma \mathbf{x}}$ where $\Sigma$ is the covariance matrix of returns.
