We consider multidimensional optimization problems formulated in the tropical mathematics setting to minimize or maximize functions defined on vectors over idempotent semifields, subject to linear equality and inequality constraints. We start with a brief overview of known tropical optimization problems and solution approaches. Furthermore, some new problems are presented with nonlinear objective functions calculated using multiplicative conjugate transposition of vectors, including problems of Chebyshev approximation, problems of approximation in the Hilbert seminorm, and pseudo-quadratic problems. To solve these problems, we apply methods based on the reduction to the solution of parametrized inequalities, matrix sparsification, and other techniques. The methods offer direct solutions represented in a compact explicit vector form ready for further analysis and straightforward computation. We conclude with a short discussion of the application of the results obtained to practical problems in location analysis, project scheduling and decision making.