What is the role of skewness in portfolio optimization?
There are many portfolio optimization paradigms that include a preference for skewness. These are generally alternatives meant to replace the modern portfolio management mean-variance framework developed by Markowitz. Skewness (or, more generally, higher moments) are only relevant in portfolio optimization if (a) assets are not normally distributed, and (b) agents have a preference for positive skewness above and beyond their preference for a higher mean and lower volatility.
One of the leading alternative paradigms which implicitly takes skewness into account is Conditional-Value-at-Risk Optimization. Attilio Meucci has a (rather long) introduction to both classical and state-of-the-art portfolio optimization. A google search for "skewness preference" also yields plenty of additional background on this topic.
A similar question on this topic is: What is the relationship between risk aversion and preference for skewness and kurtosis in portfolio optimization?
the question is very broad, Here is the brief summary of the role of all moments in portfolio optimization: expected value- the 1st moment represents the reward. All the even higher moments represent the likelihood of extreme values. Larger values for these moments indicate greater uncertainty. The odd moments represent measures of asymmetry. Skewness belongs to the second category so it represents the assymmetry. Positive skewness is desirable for investors. Skewness affects the Utility of the investor through the skewness preference. however till recently the role of higher moments in portfolio optimization was ignored in the literature. One reason for this can be the difficulties in calcualting higher moments. Becasue, to calculate the skewness or kurtosis of the portfolio the number of co-moments should be calculated.The more assets included in the portolfio, the more number of co-momnets should be calculated. On this issue I would recommend the following papers: "Improved Estimates of Higher-Order Comoments and Implications for Portfolio Selection" by Lionel Martellini and Volker Ziemann and "Optimal Hedge Fund Allocation with Improved Estimates for Coskewness and Cokurtosis Parameters" by Lionel Martellini, Asmerilda Hitaj,and Giovanni Zambruno.
Assuming you're talking about optimizing a portfolio that has options included in its investment universe.
Skewness isn't directly modeled in the optimization, although many formulations involve using implied vol as the currency numeraire. (i.e. modeling the components of skewness, instead of skewness itself)
The main impact on the optimization though is that the type of solver used changes if you want to guarantee solution uniqueness/global optimality. (Second order cone - solver).
However, this solver change is a result of simply having options in the investment universe, not from skewness per se.