Mathematical Programs with Cardinality Constraints: Reformulation by Complementarity-Type Conditions and a Regularization Method

Abstract

Optimization problems with cardinality constraints are very difficult mathematical programs which are typically solved by global techniques from discrete optimization. Here we introduce a mixed-integer formulation whose standard relaxation still has the same solutions (in the sense of global minima) as the underlying cardinality-constrained problem; the relation between the local minima is also discussed in detail. Since our reformulation is a minimization problem in continuous variables, it allows to apply ideas from that field to cardinality-constrained problems. Here, in particular, we therefore also derive suitable stationarity conditions and suggest an appropriate regularization method for the solution of optimization problems with cardinality constraints. This regularization method is shown to be globally convergent to a Mordukhovich-stationary point. Extensive numerical results are given to illustrate the behavior of this method

Complex portfolio selection via convex mixed‐integer quadratic programming: a survey

“…Constraint ensures that the whole initial budget is invested in the portfolio and in several papers (see, e.g., Burdakov et al., ) it is substituted by Constraint prevents short selling, that is, the possibility for the investor to sell financial assets not already in his/her portfolio. This financial operation is generally performed with speculative intents when the investor expects a bearish trend in the financial stock market.…”

“…Burdakov et al. () deal with the cardinality constrained portfolio problem, by introducing an NLP reformulation, whose global minima are the same of the ones of the original problem. The NLP is solved via a sequence of regularized programs (see Kanzow and Schwartz, ).…”

“…In several cases (see, e.g., Bonami and Lejeune, 2009), an additional nonrisky asset with mean μ 0 and zero variance is also considered to algorithmically derive the efficient frontier (see Section 4.8). Several papers (see, e.g., Crama and Schyns, 2003) consider an equality version for the return constraint (1b), namely μ T x = R. Constraint (1c) ensures that the whole initial budget is invested in the portfolio and in several papers (see, e.g., Burdakov et al, 2016) it is substituted by e T x ≤ 1.…”

The multiplicative weights update algorithm for mixed integer nonlinear programming: theory, applications, and limitations

“…Constraint (3.1c) ensures that the whole capital available is invested in the portfolio and in several papers (see, e.g., [38]) is substituted by…”

“…Burdakov et al [38] deal with the cardinality constrained portfolio problem, by introducing a NLP reformulation, whose global minima are the same of the ones of the original problem. The NLP is solved via a sequence of regularized programs (see [140]).…”