Max Sharpe (Markowitz) maximizes risk-adjusted return: the classic Modern Portfolio Theory optimization.

Minimum Variance finds the least volatile portfolio possible — it only uses the covariance structure, not return forecasts.

Maximum Diversification maximizes the diversification ratio (weighted average of individual volatilities over portfolio volatility). Also forecast-free.

Target Volatility maximizes expected return subject to the annualized volatility you set — useful to match a risk budget.