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It is the study of minimizing the convex functions over convex sets . Following statements of convex optimization problem :
If there is a local minimum then it is considered as global minimum.
Set of all minima is convex .
If the function has minimum then this minimum should be unique.
The usual form in which convex optimization problems are described is termed as its standard form .
It is divided into three parts : convex function , inequality constraints and equality constraints .
Convex function : f(x) : R^n -> R where x is the variable .
Inequality constraints : gi(x) <= 0, where gi is convex.
Equality constraints : It can be described in the form of equation : hi(x) =ai^Tx+bi where ai = column vector and bi = real
Convex optimization problems can be solved by following methods : Bundle method , interior point method , ellipsoid method ,
cutting plane method , subgradient method etc.
Example of convex optimized problem : least squares , linear programming , geometric programming , second order cone
programming , entropy maximization etc.
Basics of convex analysis: Convex setsfunctionsand optimization problems. Optimization theory: Least-squareslinearquadraticgeometric and semidefinite programming. Convex modeling. Duality theory. Optimality and KKT conditions. Applications in signal processingstatisticsmachine learningcontrol communicationsand design of engineering systems
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Help for complex topics like :
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- Linear quadratic optimization,Geometric programmingsemidefinite programming,Vector optimization
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Convex sets, functions, and optimization problems.
Convex analysis and theory of convex programming: optimality conditions, duality theory, theorems of alternative, and applications.
Least-squares, linear and quadratic programs, semidefinite programming, and geometric programming.
Numerical algorithms for smooth and equality constrained problems;
Interior-point methods for inequality constrained problems.
Applications to signal processing, communications, control, analog and digital circuit design, computational geometry, statistics, machine learning, and mechanical engineering.
Mathematical optimization; least-squares and linear programming; convex optimization; course goals and topics; nonlinear optimization.
Convex sets and cones; some common and important examples; operations that preserve convexity.
Convex functions; common examples; operations that preserve convexity; quasiconvex and log-convex functions.
Convex optimization problems
Convex optimization problems; linear and quadratic programs; second-order cone and semidefinite programs; quasiconvex optimization problems; vector and multicriterion optimization.
Lagrange dual function and problem; examples and applications.
Approximation and fitting
Norm approximation; regularization; robust optimization. (PDF)
Maximum likelihood and MAP estimation; detector design; experiment design.
Projection; extremal volume ellipsoids; centering; classification; placement and location problems.
Filter design and equalization
FIR filters; general and symmetric lowpass filter design; Chebyshev equalization; magnitude design via spectral factorization.
Multi-period processor speed scheduling; minimum time optimal control; grasp force optimization; optimal broadcast transmitter power allocation; phased-array antenna beamforming; optimal receiver location.
l1 methods for convex-cardinality problems
Convex-cardinality problems and examples; l1 heuristic; interpretation as relaxation.
l1 methods for convex-cardinality problems (cont.)
Total variation reconstruction; iterated re-weighted l1; rank minimization and dual spectral norm heuristic.
Stochastic programming; "certainty equivalent" problem; violation/shortfall constraints and penalties; Monte Carlo sampling methods; validation.
Chance constrained optimization
Chance constraints and percentile optimization; chance constraints for log-concave distributions; convex approximation of chance constraints.
Numerical linear algebra background
Basic linear algebra operations; factor-solve methods; sparse matrix methods.
Gradient and steepest descent methods; Newton method; self-concordance complexity analysis.
Equality constrained minimization
Elimination method; Newton method; infeasible Newton method.
Barrier method; sequential unconstrained minimization; self-concordance complexity analysis.
Disciplined convex programming and CVX
Convex optimization solvers; modeling systems; disciplined convex programming; CVX.