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Course: CAAM 236, Spring 2007
School: Monmouth IL
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(Fall EE236A 2007-08) Lecture 1 Introduction and overview linear programming example from optimal control example from combinatorial optimization history course topics software 11 Linear program (LP) n minimize j =1 n cj xj aij xj bi, j =1 n subject to i = 1, . . . , m i = 1, . . . , p cij xj = di, j =1 variables: xj R problem data: the coecients cj , aij , bi, cij , di can be solved very...

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(Fall EE236A 2007-08) Lecture 1 Introduction and overview linear programming example from optimal control example from combinatorial optimization history course topics software 11 Linear program (LP) n minimize j =1 n cj xj aij xj bi, j =1 n subject to i = 1, . . . , m i = 1, . . . , p cij xj = di, j =1 variables: xj R problem data: the coecients cj , aij , bi, cij , di can be solved very eciently (several 10,000 variables, constraints) widely available high-quality software extensive, useful theory (optimality conditions, sensitivity analysis, . . . ) Introduction and overview 12 Example: open-loop control problem single-input/single-output system (with input u, output y ) y (t) = h0u(t) + h1u(t 1) + h2u(t 2) + h3u(t 3) + output tracking problem: minimize deviation from desired output ydes(t) t=0,...,N max |y (t) ydes(t)| subject to input amplitude and slew rate constraints: |u(t)| U, |u(t + 1) u(t)| S variables: u(0), . . . , u(M ) (with u(t) = 0 for t < 0, t > M ) solution: can be formulated as an LP, hence easily solved (more later) Introduction and overview 13 example step response (s(t) = ht + + h0) and desired output: step response ydes(t) 1 1 0 0 0 100 200 1 0 100 200 amplitude and slew rate constraint on u: |u(t)| 1.1, |u(t) u(t 1)| 0.25 Introduction and overview 14 optimal solution (computed via linear programming) input u(t) u(t) u(t 1) 0.25 1.1 0.0 0.00 1.1 0 100 200 0.25 0 100 200 output and desired output 1 0 1 0 100 200 Introduction and overview 15 Example: assignment problem match N people to N tasks each person is assigned one task; each task assigned to one person cost of assigning person i to task j is aij combinatorial formulation minimize subject to N i,j =1 aij xij N i=1 xij = 1, N j =1 xij j = 1, . . . , N i = 1, . . . , N i, j = 1, . . . , N = 1, xij {0, 1}, variable xij = 1 if person i is assigned to task j ; xij = 0 otherwise N ! possible assignments, i.e., too many to solve by enumeration Introduction and overview programming 16 linear formulation minimize subject to N i,j =1 aij xij N i=1 xij = 1, N j =1 xij j = 1, . . . , N i = 1, . . . , N i, j = 1, . . . , N = 1, 0 xij 1, we have relaxed the constraints xij {0, 1} it can be shown that at the optimum xij {0, 1} (see later) hence, can solve (this particular) combinatorial problem eciently (via LP or specialized methods) tractable combinatorial optimization problems usually have ecient equivalent LP formulations Introduction and overview 17 Brief history 1940s (Dantzig,Kantorovich, Koopmans, von Neumann, . . . ): foundations of LP, motivated by economic and logistics problems 1947 (Dantzig): simplex algorithm 1950s60s applications in other elds (structural optimization, control theory, lter design, . . . ) 1979 (Khachiyan) ellipsoid algorithm: more ecient (polynomial-time) than simplex in worst case, much slower in practice 1984 (Karmarkar): projective (interior-point) algorithm: polynomial-time worst-case complexity, and ecient in practice 1984today. variations of interior-point methods (improved complexity or eciency in practice), software for large-scale problems Introduction and overview 18 Course outline the linear programming problem linear inequalities, geometry of linear programming applications signal processing, control, structural optimization . . . duality algorithms the simplex algorithm, interior-point algorithms large-scale linear programming techniques for LPs with special structure integer linear programming introduction, some basic techniques Introduction and overview 19 Software solvers: solve LPs described in some standard form modeling tools: accept a problem in a simpler, more intuitive, notation and convert it to the standard form required by solvers software for this course (see class website) platforms: Matlab, Octave, Python solvers: linprog (Matlab Optimization Toolbox), lp236a.m (Matlab/Octave), MOSEK (Matlab), CVXOPT (Python) modeling tools: CVX (Matlab), YALMIP (Matlab), CVXOPT (Python) Introduction and overview 110
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University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808
University of Florida - CHM - 0808