Convex programming

In summary, the optimum of a nonlinear programming problem is, in general, not at an extreme point of the feasible region and may not even be on the boundary. Also, the problem may have local optima distinct from the global optimum. These properties are direct consequences of nonlinearity. A class of nonlinear problems can be defined, however, that are guaranteed to be free of distinct local optima. They are called convex programming problems and are considered in the following section. 

An important result in mathematical programming evolves from the concept of convexity. For the nonlinear programming problem called the convex programming problem... 

Although convexity is desirable, many real-world problems turn out to be non-convex. In addition, there is no simple way to demonstrate that a nonlinear problem is a convex problem for all feasible points. Why, then is convex programming studied The main reasons are... 

Y. Nesterov and A. S. Nemirovskii, Interior Point Polynomial Method in Convex Programming Theory and Applications, SIAM, Philadelphia, 1993. 

The energy and spectral optimization problems are convex programs so when there are multiple solutions the solution sets form a convex set. The following corollary characterizes how these convex sets of solutions relate to solutions of the Euler equation. In the formulation of this corollary we use the notion of optimal gap Ao—the gap achieved by optimal P and S. The optimal gap is a characteristic of the energy problem, depending only on H and S. 

S.I. Zukhovitskii and L.I. Avdeeva, Linear and Convex Programming, Nauka, Moscow, 1967 (in Russian). 

Note that when setting the list x , j = 1 the authors deviate from the classical Gibbs definition, understanding by the system components not individual substances but their quantities contained in a certain phase. For example, if the water in reaction mixture is in gaseous and condensed phases, its corresponding phase concentrations represent different parameters of the studied system. Such expansion of the space of variables of the problem solved facilitates its reduction to the problems of convex programming (CP). 

If possible, reformulate the MINLP as a convex programming problem. 

Kelley, J. E.. Jr.. The Cutting-Plane Method for Solving Convex Programs, J. SIAM 8, 703-712 (I960). 

Ceria S. and Soares J. 1999. Convex programming for disjunctive optimization. Math. Program., 86(3), 595-614. 

Kelley J.E. Jr. 1960. The cutting-plane method for solving convex programs, J. SIAM, 8, 703. 

Stubbs R. and Mehrotra S. 1999. A branch-and-cut method for 0-1 mixed convex programming. Mathematical Programming, 86(3), 515-532. 

If H is positive semidefinite, then problem (QP) is a convex program. Thus, the KKT conditions are sufficient in this case, and any solution to this system will yield a global optimal solution to (QP). When H is indefinite, then local optimal solutions which are not global optimal solutions may... 

RockafeUar, R. T. (1973), The Multiplier Method of Hestenes and Powell Applied to Convex Programming, Journal of Optimization Theory and Applications, Vol. 12, pp. 555-562. 

If X and ffx), i = 1, 2,. . . , q have some special stracture, then more efficient computational techniques are available. For instance, if X is a convex set and each ffx) is concave, then we first have q concave programming and then a convex programming (because r(f(x) p) is convex under the assumptions). If X is a polyhedron defined by a system of Unear inequalities and each ffx) is hnear, then the ideal points y can be found by q simple linear programming problems. Furthermore, the compromise solutions of y and y can be found by a Unear programming problem (the other compromise solutions, y, 1 < p < >, can be found by convex programming) (Yu 1985). 

Our edgelit convex programmed flashing safety border will demand attention... 

Soyster AL (1973) Convex programming with set-inclusive constraints and applications to inexact linear programming. OperRes 21 1154-1157... 

Nesterov Y, Nemirovskii AS (1993) Interior point polynomial method in convex programming theory and applications. SIAM, Philadelphia... 

The concept of convexity is fundamental in optimization. Many practical problems possess this property, which generally makes them easier to solve both in theory and practice. If the objective function in the optimization problem (1) and the feasible region are both convex, then any local solution of the problem is in fact a global solution. The term convex programming is used to describe a special case of the general constrained optimization problem in which ... 

Convex programming refers to a minimization problem with a convex objective and a convex constraint set. For example, if/and g are convex functions, then... 

The performed analysis of problems solved by using MEIS has shown the possibilities for their reduction to convex programming (CP) problems in many important cases. Such reduction is often associated with approximation of dependences among variables. There are cases of multivalued solutions to the formulated CP problems, when the linear objective function is parallel to one of the linear part of set D y). Naturally the problems with non-convex objective functions or non-convex attainability sets became irreducible to CP. Non-convexity of the latter can occur at setting kinetic constraints by a system of linear inequalities, p>art of which is specified not for the whole region D (y), but its individual zones. 

Shirkalin, LA. (1997). Solution of Convex Programming Problem with Large Scatter cf Variable Values, SEI AS USSR, Irkutsk (in Russian). 

The first part briefly introduces the stochastic expected value programming theory. With the discussing on the expected value model which is a convex programming, Theorem 4.2 is put forward and proved. Furthermore we get the conclusion that if the expected value model is a convex programming and there exists an optimal solution, then any local optimal solution will be the global optimal solution. 

Thus it can be seen that + (1 — is a feasible solution to (4.6) and the feasible set is convex. Therefore the expected value model (4.6) is a convex programming model. This completes the proof. ... 

Theorem 4.2 For the convex programming (4.11), assume D is a feasible region, then ... 

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