Convex relaxation nonconvex optimization

  • What is convex optimization vs nonconvex optimization?

    The convex optimization problem refers to those optimization problems which have only one extremum point (minimum/maximum), but the non-convex optimization problems have more than one extremum point..

  • What is convex relaxation?

    Relaxation denotes the technique of simply dropping certain constraints from the overall optimization problem.
    Convex relaxation means that upon relaxation the problem becomes convex..

1.2 Convex Relaxation and Engineering Problems The optimal value of the convex relaxation is a lower bound for the optimal value of the original problem. This lower bound can play important roles in solving and analyzing nonconvex optimization problems, in cases we need an actual solution to the original problem.
1.2 Convex Relaxation and Engineering Problems The optimal value of the convex relaxation is a lower bound for the optimal value of the original problem. This lower bound can play important roles in solving and analyzing nonconvex optimization problems, in cases we need an actual solution to the original problem.
In this dissertation, we ana- lyze two prominent ways of obtaining convex relaxations: Lagrangian duality and semidefinite programming. The difference between 

What is non convex optimization?

Non-convex optimization This inequality simply requires that the gradient grows faster than a linear function as we move away from the optimal function value

All convex functions rates apply

Issue: dealing with saddle points Issue: Exponential number of saddle points

Which convex relaxation methods satisfy Assumption 3?

Several established convex relaxation approaches produce functions \ ( (\mathbf {f}^\mathrm {cv},\mathbf {f}^\mathrm {cc})\) that satisfy Assumption 3, including certain affine relaxations, the \ (\alpha \) BB relaxations [ 2 ], the McCormick relaxations [ 40 ], or the pointwise tightest among certain multiple relaxations defined as follows


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