one of the significant strategies of the power systems is economic dispatch (ed) problem, which is defined as the optimal generation of power units to produce energy at the lowest cost by fulfilling the demand within several limits. the undeniable impacts of ramp rate limits, valve loading, prohibited operating zone, spinning reserve and multi-fuel option on the economic dispatch of practical p...

Optimization problems are not only formed into a linear programming but also nonlinear programming. In real life, often decision variables restricted on integer. Hence, came the nonlinear programming. One particular form of nonlinear programming is a convex quadratic programming which form the objective function is quadratic and convex and linear constraint functions. In this research designed ...

In this paper, an integrated machine scheduling withits due date setting problem has been considered. It is assumed that the machine is subject to some kind of random unavailability. Due dates should be set in an attractive and reliable manner, implying that they should be short and possible to be met. To this end, first, long due dates are penalized in the objective function. Then, for each cu...

Non-convex sparsity-inducing penalties have recently received considerable attentions in sparse learning. Recent theoretical investigations have demonstrated their superiority over the convex counterparts in several sparse learning settings. However, solving the non-convex optimization problems associated with non-convex penalties remains a big challenge. A commonly used approach is the Multi-S...

As a first-order method, the augmented Lagrangian method (ALM) is a benchmark solver for linearly constrained convex programming, and in practice some semi-definite proximal terms are often added to its primal variable's subproblem to make it more implementable. In this paper, we propose an accelerated PALM with indefinite proximal regularization (PALM-IPR) for convex programming with linear co...

Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient primal-dual interior-point methods (IPMs) and various first order approaches for the solution of ...

In the previous lecture, in the first place, we talked about duality for general non-convex optimization. And we know that dual functions are always convex. The dual of the dual problem is also convex. In applications, the dual of the dual can be used as the convex relaxation of the primal (and we will see this explicitly in the next lecture). Then, we covered the concepts of weak duality and s...

In this paper we provide estimates of the rates of convergence of monotone approximation schemes for non-convex equations in one spacedimension. The equations under consideration are the degenerate elliptic Isaacs equations with x-depending coefficients, and the results applies in particular to finite difference methods and control schemes based on the dynamic programming principle. Recently, K...