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Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization which may be considered a quasi-Newton method.SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable, but not necessarily convex.
In the EQP phase of SLQP, the search direction of the step is obtained by solving the following equality-constrained quadratic program: + + (,,).. + = + =Note that the term () in the objective functions above may be left out for the minimization problems, since it is constant.
The algorithm starts with an initial estimate of the optimal value, , and proceeds iteratively to refine that estimate with a sequence of better estimates ,, ….The derivatives of the function := are used as a key driver of the algorithm to identify the direction of steepest descent, and also to form an estimate of the Hessian matrix (second derivative) of ().
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Quadratic programming is particularly simple when Q is positive definite and there are only equality constraints; specifically, the solution process is linear. By using Lagrange multipliers and seeking the extremum of the Lagrangian, it may be readily shown that the solution to the equality constrained problem
There are two main relaxations of QCQP: using semidefinite programming (SDP), and using the reformulation-linearization technique (RLT). For some classes of QCQP problems (precisely, QCQPs with zero diagonal elements in the data matrices), second-order cone programming (SOCP) and linear programming (LP) relaxations providing the same objective value as the SDP relaxation are available.
Which one is best with respect to the number of function calls depends on the problem itself. Methods that evaluate Hessians (or approximate Hessians, using finite differences): Newton's method; Sequential quadratic programming: A Newton-based method for small-medium scale constrained problems. Some versions can handle large-dimensional problems.
A linear programming problem is one in which we wish to maximize or minimize a linear objective function of real variables over a polytope.In semidefinite programming, we instead use real-valued vectors and are allowed to take the dot product of vectors; nonnegativity constraints on real variables in LP (linear programming) are replaced by semidefiniteness constraints on matrix variables in ...