{"id":2506,"date":"2023-06-28T14:53:59","date_gmt":"2023-06-28T14:53:59","guid":{"rendered":"https:\/\/nag.com\/?page_id=2506"},"modified":"2024-02-21T14:18:05","modified_gmt":"2024-02-21T14:18:05","slug":"solving-convex-problems-with-second-order-cone-programming-socp","status":"publish","type":"page","link":"https:\/\/nag.com\/solving-convex-problems-with-second-order-cone-programming-socp\/","title":{"rendered":"Solving Convex Problems with Second-order Cone Programming (SOCP)"},"content":{"rendered":"\n
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Solving Convex Problems with Second-order Cone Programming (SOCP)<\/h1>\n \n
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Second-order cone programming (SOCP) offers a robust and efficient way of solving several types of convex problems, including convex quadratically constrained quadratic programming (QCQP) problems. That is the reason why it can be highly utilized in a broad range of\u2000fields, like \u2000finance. At Mark 29.3 of the n<\/i><\/span>AG Library, n<\/i><\/span>AG introduces a performance update to the SOCP solver (nag_opt_handle_solve_socp_ipm<\/a>) which has been shown to be particularly effective for portfolio optimization.<\/div>\n
\u00a0<\/div>\n
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1. SOCP in the n<\/i><\/span>AG Library Introduction<\/h3>\n

Second-order cone programming is one of the main methods for solving QCQP problems, which arises in applications such as portfolio construction. The standard form convex QCQP problem is:<\/span><\/p>\n<\/div>\n<\/div>\n <\/div>\n <\/div>\n<\/div>\n\n\n\n minimize<\/mtext><\/mstyle><\/mrow>x<\/mi> \u2208<\/mo> \u211d<\/mi><\/mrow>n<\/mi><\/mrow><\/msup><\/mrow><\/munder><\/mtd>1<\/mn><\/mrow>\n2<\/mn><\/mrow><\/mfrac>x<\/mi><\/mrow>T<\/mi><\/mrow><\/msup>Q<\/mi><\/mrow>\n0<\/mn><\/mrow><\/msub>x<\/mi> +<\/mo> r<\/mi><\/mrow>0<\/mn><\/mrow>T<\/mi><\/mrow><\/msubsup>x<\/mi> <\/mtd>\n<\/mtr> <\/mtr> subject to<\/mtext><\/mstyle> <\/mtd>1<\/mn><\/mrow>\n2<\/mn><\/mrow><\/mfrac>x<\/mi><\/mrow>T<\/mi><\/mrow><\/msup>Q<\/mi><\/mrow>\ni<\/mi><\/mrow><\/msub>x<\/mi> +<\/mo> r<\/mi><\/mrow>i<\/mi><\/mrow>T<\/mi><\/mrow><\/msubsup>x<\/mi> +<\/mo> s<\/mi><\/mrow>\ni<\/mi><\/mrow><\/msub> \u2264<\/mo> 0<\/mn><\/mspace>for<\/mi><\/mstyle><\/mspace>i<\/mi> =<\/mo> 1<\/mn>,<\/mo>\u2026<\/mi>,<\/mo>m<\/mi>,<\/mo><\/mtd>\n<\/mtr> <\/mtr> <\/mtd>Ax<\/mi> =<\/mo> b<\/mi>,<\/mo>\n\n\n
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where \\(Q_{o},…, Q_{m} \\) are positive semidefinite \\(n\\) x \\(n\\) matrices.<\/p>\n

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Figure\u00a01:\u00a0<\/span>Feasible region of an SOCP problem with three variables.<\/span><\/p>\n

The feasible region of an SOCP problem (which may, for instance, be a transformed QCQP problem) can be seen in Figure\u00a01. The feasible region here is the intersection of the green polyhedron and a cone. The curve represents the added quadratic information which makes SOCP more complicated, but at the same time more powerful than LP.<\/p>\n

In terms of the implementation of n<\/i><\/span>AG\u2019s SOCP solver, a path-following homogeneous self-dual algorithm is used. This is both robust and efficient and can detect infeasibility and unboundedness of the model. Also, n<\/i><\/span>AG exploits the sparsity pattern of the problem thoroughly which makes the solver efficient, especially on large and sparse problems. For more details on implementation, see the n<\/i><\/span>AG Library routine documentation for n<\/i><\/span>AG SOCP solver nag_opt_handle_solve_socp_ipm<\/span><\/a>\u00a0[1]<\/span>.<\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n

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2. <\/span>Usage of SOCP in Portfolio Optimization<\/h3>\n

It is rare that a second-order cone (SOC) would appear naturally in the model, yet through a model reformulation, many portfolio optimization problems and constraints can be handled by SOCP. We discuss several of these below, for more examples see\u00a0[2]<\/span>.<\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n

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2.1\u00a0<\/span>Optimization with norms<\/h4>\n

The leverage constraint:<\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n\n \u2225<\/mo>x<\/mi>\u2225<\/mtext><\/mrow>1<\/mn><\/mrow><\/msub> =<\/mo> \u2211<\/mo>\n <\/mrow>i<\/mi>=<\/mo>1<\/mn><\/mrow>n<\/mi><\/mrow><\/munderover>|<\/mo>x<\/mi><\/mrow>\ni<\/mi><\/mrow><\/msub>|<\/mo>\u2264<\/mo> L<\/mi><\/mspace>\u27fa<\/mo><\/mspace>\u2211<\/mo>\n <\/mrow>i<\/mi>=<\/mo>1<\/mn><\/mrow>n<\/mi><\/mrow><\/munderover>s<\/mi><\/mrow>\ni<\/mi><\/mrow><\/msub> \u2264<\/mo> L<\/mi>,<\/mo> <\/mi> (<\/mo>s<\/mi><\/mrow>i<\/mi><\/mrow><\/msub>,<\/mo>x<\/mi><\/mrow>i<\/mi><\/mrow><\/msub><\/mrow>)<\/mo><\/mrow> \u2208<\/mo>K<\/mi><\/mstyle><\/mrow>q<\/mi><\/mrow>2<\/mn><\/mrow><\/msubsup>,<\/mo> i<\/mi> =<\/mo> 1<\/mn>,<\/mo>\u2026<\/mo>,<\/mo>n<\/mi>.<\/mo>\n<\/mrow><\/math>\n\n\n
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\u00a0<\/p>\n

The above norm constraint can be viewed as a special case of general\u00a0p<\/span>-norm constraint on a vector x<\/mi> \u2208<\/mo> \u211d<\/mi><\/mrow>n<\/mi><\/mrow><\/msup><\/mrow><\/math> defined as <\/mrow><\/math><\/p>\n

\u00a0<\/p>\n

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\u00a0<\/mi><\/mrow><\/math><\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n\n <\/mstyle>\u2225<\/mo>x<\/mi>\u2225<\/mtext><\/mrow>p<\/mi><\/mrow><\/msub> :<\/mo>=<\/mo>(<\/mo>\u2211<\/mo>\n <\/mrow>i<\/mi>=<\/mo>1<\/mn><\/mrow>n<\/mi><\/mrow><\/munderover>|<\/mo>x<\/mi><\/mrow>\ni<\/mi><\/mrow><\/msub>|<\/mtext><\/mrow>p<\/mi><\/mrow><\/msup><\/mrow>)<\/mo><\/mrow><\/mrow><\/mspace>1<\/mn>\u2215<\/mo>p<\/mi><\/mrow><\/msup> \u2264<\/mo> t<\/mi>,<\/mo>\n<\/mrow><\/math>\n\n\n
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(1)<\/p>\n

Where \\(p = l \/ m \\) is a positive rational number with \\( l \u2265 m \\). Inequality constraints (1) have SOC representation since they are equivalent to inequalities involving rational powers.<\/p>\n

2.2\u00a0<\/span>Quadratic constraint<\/h4>\n

The tracking error constraint<\/p>\n

\u00a0<\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n\n <\/mstyle>1<\/mn><\/mrow>\n2<\/mn><\/mrow><\/mfrac>x<\/mi><\/mrow>T<\/mi><\/mrow><\/msup>Px<\/mi> +<\/mo> q<\/mi><\/mrow>T<\/mi><\/mrow><\/msup>x<\/mi> +<\/mo> r<\/mi> \u2264<\/mo> 0<\/mn>\n<\/mrow><\/math>\n\n\n
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(2)<\/p>\n

where \\(P\\) \u2208 \\(S^n\\) is a positive semidefinite matrix is common in portfolio optimization. Using routines handle_set_qconstr<\/span><\/a>\u00a0(e04rs<\/a>) and\u00a0handle_set_qconstr_fac<\/span><\/a>\u00a0(e04rt<\/a>) introduced at Mark 27.1 of the n<\/i><\/span>AG Library, users can easily define quadratic objective functions and\/or constraints and seamlessly integrate them with other constraints. Then what will happen under the hood is as follows. Assume the matrix \\( P \\) has factorization \\( P = F^TF \\), (2) has an equivalent SOC representation as<\/mo><\/p>\n

\u00a0<\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n\n t<\/mi> +<\/mo> q<\/mi><\/mrow>T<\/mi><\/mrow><\/msup>x<\/mi> +<\/mo> r<\/mi> =<\/mo> 0<\/mn>,<\/mo> <\/mi> (<\/mo>t<\/mi>,<\/mo>1<\/mn>,<\/mo>Fx<\/mi><\/mrow>)<\/mo><\/mrow> \u2208<\/mo>K<\/mi><\/mrow>\nr<\/mi><\/mrow>n<\/mi>+<\/mo>2<\/mn><\/mrow><\/msubsup>\n<\/mrow><\/math>\n\n\n
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\u00a0<\/p>\n

since \\( (1 \/ 2)x^TPx \u2264\u00a0 t \\) is equivalent to \\( || Fx||^2_{2} \u2264 2t\\). Thus, adding auxiliary variables \\( y = Fx \\) will transform the quadratic constraint into a standard cone constraint.
<\/mn><\/mrow><\/msubsup><\/mrow><\/math><\/p>\n

For more details, see our\u00a0mini article on QCQP<\/a>\u00a0and\u00a0the python examples on portfolio optimization<\/a>\u00a0involving quadratic functions.<\/p>\n

2.3\u00a0<\/span>More second-order cone representable functions<\/h4>\n

We have mentioned important SOC-representable functions above yet many more problems and constraints from portfolio optimization are SOC-representable. We list some below.<\/p>\n