{"id":1056,"date":"2023-05-25T08:50:49","date_gmt":"2023-05-25T08:50:49","guid":{"rendered":"https:\/\/nag.com\/?page_id=1056"},"modified":"2023-07-07T14:56:50","modified_gmt":"2023-07-07T14:56:50","slug":"solving-quadratically-constrained-quadratic-programming-qcqp-problems","status":"publish","type":"page","link":"https:\/\/nag.com\/solving-quadratically-constrained-quadratic-programming-qcqp-problems\/","title":{"rendered":"Solving quadratically constrained quadratic programming (QCQP) problems"},"content":{"rendered":"\n
\n
\n
\n
\n
\n
\n \n

Solving quadratically constrained quadratic programming (QCQP) problems<\/h1>\n \n
<\/div>\n\n \n <\/div>\n <\/div>\n <\/div>\n <\/div>\n <\/div>\n<\/div>\n\n\n\n\n
\n
\n
\n \n \n
\n
\n

Quadratic functions are a powerful modelling construct in mathematical programming and appear in various disciplines such as statistics, machine learning (Lasso regression), finance (portfolio optimization), engineering (OPF) and control theory. At Mark 27.1 of the n<\/i><\/span>AG Library, n<\/i><\/span>AG introduced two new additions to the n<\/i><\/span>AG Optimization Modelling Suite to help users easily define quadratic objective functions and\/or constraints, seamlessly integrate them with other constraints and solve the resulting problems using compatible solvers without the need of a reformulation or any extra effort.<\/p>\n

Formally, a QCQP problem can be written in its pure form with only quadratic functions in both objective and constraints as follows:<\/p>\n <\/div>\n <\/div>\n\n \n \n

\n
\n

\"\"<\/p>\n <\/div>\n <\/div>\n\n \n \n <\/div>\n <\/div>\n<\/div>\n\n\n minimize<\/mtext><\/mstyle><\/mrow>x<\/mi> \u2208<\/mo>\u211c<\/mi><\/mrow>n<\/mi><\/mrow><\/msup><\/mrow><\/munder><\/mtd>1<\/mn><\/mrow> 2<\/mn><\/mrow><\/mfrac>x<\/mi><\/mrow>T<\/mi><\/mrow><\/msup>Q<\/mi><\/mrow> 0<\/mn><\/mrow><\/msub>x<\/mi> +<\/mo> r<\/mi><\/mrow>0<\/mn><\/mrow>T<\/mi><\/mrow><\/msubsup>x<\/mi> +<\/mo> s<\/mi><\/mrow> 0<\/mn><\/mrow><\/msub> <\/mtd> <\/mtr> <\/mtd><\/mtd><\/mtr> subject to<\/mtext><\/mstyle> <\/mtd>1<\/mn><\/mrow> 2<\/mn><\/mrow><\/mfrac>x<\/mi><\/mrow>T<\/mi><\/mrow><\/msup>Q<\/mi><\/mrow> i<\/mi><\/mrow><\/msub>x<\/mi> +<\/mo> r<\/mi><\/mrow>i<\/mi><\/mrow>T<\/mi><\/mrow><\/msubsup>x<\/mi> +<\/mo> s<\/mi><\/mrow> i<\/mi><\/mrow><\/msub> \u2264<\/mo> 0<\/mn>,<\/mo><\/mspace>i<\/mi> =<\/mo> 1<\/mn>,<\/mo>\u2026<\/mo>,<\/mo>p<\/mi>,<\/mo><\/mtd><\/mtr><\/mtable> <\/math>\n\n\n\n
<\/div>\n\n
\n
\n
\n

where \\(Q_{i} \\in \\mathbb{R}^{n \\times n}, i = 0, \\ldots, p\\) are symmetric matrices, \\( r_{i} \\in \\mathbb{R}^{n}, i = 0, \\ldots, p \\) are vectors and \\(s_i\\) scalars. However, in practice it will often be stated with linear constraints and simple bounds as well. The two new routines are\u00a0handle<\/span>_set<\/span>_qconstr<\/span><\/a>\u00a0<\/span>(e04rs<\/a><\/span>) which defines quadratic functions in\u00a0assembled form\u00a0<\/span>as above and handle<\/span>_set<\/span>_qconstr<\/span>_fac<\/span><\/a>\u00a0<\/span>(e04rt<\/a><\/span>) which uses\u00a0factored form \\( \\frac{1}{2} x^{T} F_{i}^{T} F_{i} x + r_{i}^{T} x + s_{i} \\). The latter form is useful in many applications where the factors \\(F_i\\) appear naturally, such as in data fitting \\( \\| Fx – b \\|_{2} \\).
<\/span><\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n

<\/div>\n\n\n\n
\n
\n
\n \n \n
\n
\n

\"\"<\/p>\n <\/div>\n <\/div>\n\n \n \n

\n
\n

The key question is if the problem is\u00a0convex\u00a0<\/span>or\u00a0non-convex\u00a0<\/span>as it determines if the problem can be solved via conic optimization (second-order cone programming, SOCP) or only by generic nonlinear programming (NLP). The problem is convex if all \\( Q_{i}, i = 0, \\ldots, p \\) are positive semidefinite (the factored form is positive semidefinite by definition).<\/p>\n <\/div>\n <\/div>\n\n \n \n <\/div>\n <\/div>\n<\/div>\n\n\n

<\/div>\n\n
\n
\n
\n

QCQP problems were solvable even without the new additions in this release but a nontrivial knowledge of\u00a0reformulation techniques<\/a><\/span>\u00a0was required if an SOCP solver was to be used or callbacks covering the quadratic functions had to be introduced in the general case. To remove the unnecessary burden from practitioners, the new routines provide the quadratic data directly to the solvers with an appropriate automatic reformulation as is required by the chosen solver from the Suite.<\/span><\/p>\n

\u00a0<\/p>\n