In this thesis, we mainly concentrate on the mean-variance portfolio selection problems with cardinality constraint and/or quantity constraints. These combinatorial problems are NP-hard in general. The first model is the Sharpe ratio portfolio selection problem (2.4) which is a single-period assets selection optimization problem maximizing the Sharpe ratio of a portfolio containing exactly k stocks which are selected from n stocks in the market, and shorting is allowed in this model. We provide an approximation solution for the Sharpe ratio portfolio optimization problem with a worst-case performance guarantee. In the second model, we consider the portfolio selection problem which takes into account both the cardinality constraint and the quantity constraint, i.e., limiting the number of assets and the minimal and maximal shares of each individual asset in the portfolio, respectively, which is reformulated as mixed 0-1 conic programming. In the third model, we consider the random portfolio selection scheme, i.e., we randomly select some stocks into our portfolio either with constant probability or by controlling the probability. In the last model, we assume that investors only would like to either invest in an asset with a substantial amount (represented by some threshold value) or discard it. With the help of the SDP relaxation, a screening algorithm, and a randomized rounding procedure, we find approximative solutions whose worst-case guaranteed performance bound is O( m3). Branch-and-bound method is also considered to find the exact optimal solution for this model.;Keywords. portfolio selection, cardinality, quantity, threshold, SDP relaxation, random rounding procedure, mixed 0-1 conic programming.
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