# characterization of local solutions useful for solving reverse convex programs

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The Physical Object ID Numbers Statement Richard J. Hillestad. Series Rand Corporation : Paper -- P-5066. Pagination iii, 20 p ; Open Library OL16457650M

A contribution to mathematical programming theory, describing a solution characterization of local solutions useful for solving reverse convex programs book for a class of nonconvex programs defined by constraints and objectives having convexity which is the reverse of that required for a convex problem.

A Characterization of Local Solutions Useful for Solving Reverse Convex Problems. by It is also shown that. Get this from a library. A characterization of local solutions useful for solving reverse convex programs. [Richard J Hillestad; Rand Corporation.]. Abstract. The sequential linear programming (SLP) method for solving nonlinear problems was introduced in the s.

Many papers that attempted to use SLP reported poor performance and convergence issues.

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We found that nonlinear programs with reverse convex constraints, which are the most difficult nonlinear programs with many local optima, are solved (heuristically) very well by : Zvi Drezner, Pawel Jan Kalczynski. It is well known that each convex function \(f:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}\) is supremally generated by affine functions.

More precisely, each convex function \(f:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}\) is the upper envelope of its affine minorants. In this paper, we propose an algorithm for solving reverse convex programming problems by using such Author: J.

Sadeghi, H. Mohebi. Stephen E. Jacobsen. Request full-text PDF. Reverse convex programs generally have disconnected feasible regions. Basic solutions are defined and properties of the latter and of the convex hull of the feasible region are derived.

Solution procedures are discussed and a cutting plane algorithm is developed. Some of them can be used to solve general reverse convex programming. Global solution locating is to identify the location of the solution.

The linear relaxation method is used to obtain the lower bound of the optimum of the primal programming, and in this paper the relaxed programming is a kind of linear programming, which can be solved by standard simplex algorithm.

A Finite Algorithm for Solving Linear Programs with an Additional Reverse Convex Constraint. Authors; Characterization of basic solutions for a class of nonconvex program, J. Optim. Theory and Applications 15 Linear program with an additional reverse-convex constraint, Appl.

Math. Optim., 6(), – 2 Convex sets Let c1 be a vector in the plane de ned by a1 and a2, and orthogonal to example, we can take c1 = a1 aT 1 a2 ka2k2 2 a2: Then x2 S2 if and only if j cT 1 a1j c T 1 x jc T 1 a1j: Similarly, let c2 be a vector in the plane de ned by a1 and a2, and orthogonal to a1, e.g., c2 = a2 aT 2 a1 ka1k2 2 a1: Then x2 S3 if and only if j cT 2 a2j c T 2 x jc T 2 a2j: Putting it all.

### Details characterization of local solutions useful for solving reverse convex programs FB2

This paper characterizes the solution of the second-order quadratic problem for the nondecreasing sequence {x,}"^1' e R"+r by a nonlinear but convex optimization problem, which provides not only a concrete, com- putable characterization, but also enables us to use routinely available optimization techniques (e.g., [14, 22]) for determining an.

Sen S, Sherali HD () Nondifferentiable reverse convex programs and facetial convexity cuts via a disjunctive characterization.

Math Program –. Nagai, H. and T. Kuno, "A conical branch-and-bound algorithm for a class of reverse convex programs", Proceedings of the Fourth International Conference on Nonlinear Analysis and Convex Analysis.

A constraintg(x)⩾0 is said to be a reverse convex constraint if the functiong is continuous and strictly quasi-convex. The feasible regions for linear programs with an additional reverse convex constraint are generally non-convex and disconnected.

It is shown that the convex hull of the feasible region is a convex polytope and, as a result, there is an optimal solution on an edge of the. 2 Reverse convex problems programs .

All these various expressions are used to describe the constraint g(x)≥0 that creates diﬃculties and destroys the convexity of the problem. Throughout this paper we will call the problem “the reverse convex.” A good deal of literature exists on methods for solving problem (RP).

Our focus in. Sequential convex programming (SCP) • a local optimization method for nonconvex problems that leverages convex optimization – convex portions of a problem are handled ‘exactly’ and eﬃciently • SCP is a heuristic – it can fail to ﬁnd optimal (or even feasible) point –.

In particular, in , a solution in terms of the Hamilton–Jacobi inequalities (HJIs) is proposed. It is well known that solving the HJIs is hard because there are no computational tools available for solving them. Meanwhile, a convex solution to the filter problem has been given in  through the S-procedure method.

In this paper, the. [Show full abstract] Tuy's method for convex programs with an additional reverse convex constraint to solve the converted problem.

By this way, we construct an algorithm which reduces the problem. THE computational methods of linear programming can be naturally generalised for solving convex programming problems.

Suppose it is necessary to find the minimum /o (u) subject to the conditions fi (u) convex compact set of some linear space,/o (u), fn (u) are convex functions defined on U. An inequality g{x) 2i 0 is often said to be a reverse convex constraint if the function g is conti­ nuous and convex.

The feasible regions for linear program with an additional reverse convex constraint are generally non-convex and disconnected. In this paper a convergent algorithm for solving such a linear problem is proposed.

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The. On solving general reverse convex programming problems by a sequence of linear programs and line searches Annals of Operations Research, Vol. 25, No. 1 A note on adapting methods for continuous global optimization to the discrete case. Extended reverse-convex programming: an approximate enumeration approach to global optimization Gene A.

Bunin1 Although obtaining a single solution candidate so that an optimal solution x∗ to any regular RCP problem can be obtained by solving the convex problem x.

() On the stability of general convex programs under slater’s condition and primal solution boundedness. Optimization() Convex programs with an additional constraint on the product of several convex functions. “This is a very pleasant text on global optimization problems, concentrating on those problems that have some kind of convexity property.

It describes itself as a Ph.D. level text, but in fact it is easy to read and develops all the necessary background, so it could be used by any sufficiently-motivated student. Another necessary condition for (unconstrained) local optimality of a point x was r2f(x) 0.

Note that a convex function automatically passes this test. 3 Strict convexity and uniqueness of optimal solutions Characterization of Strict Convexity Recall that a fuction f: Rn!Ris strictly convex if 8x;y;x6=y;8 2(0;1), f(x+ (1)y). Solving Optimization Problems Subject to a Budget Constraint with Economies of Scale A Characterization of Local Solutions Useful for Solving Reverse Convex Problems.

A Cutting Plane Algorithm for Problems Containing Convex and Reverse Convex Constraints. The first two characterizations can be determined in polynomial time by solving m linear programs for (i) and m convex quadratic programs for (ii), where m is the number of constraints defining.

In book: Mathematical Programming at Oberwolfach II (pp) and in the final stage we further improve the solution by solving a series of subproblems. we suppose that the convex hull of. • geometric programming • generalized inequality constraints • semideﬁnite programming • vector optimization = x3−3x, p⋆ = −∞, local optimum at x = 1 Convex optimization problems 4–3.

Implicit constraints two problems are (informally) equivalent if the solution of one is readily obtained from the solution of the. The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm.

The facial structure of the atomic norm ball carries a number of favorable properties that are useful for re-covering simple models, and an analysis of the underlying convex geometry provides.

Solving Convex Programs by Random Walks that the set contains x or returns a halfspace that separates the set from x. For the special case of linear programming, the oracle simply checks if the query point satisﬁes all the constraints of the linear program, and if not, reports a violated.

The algorithm is proven to converge to a global solution of the nonconvex program. We discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs, for which the algorithm has been implemented.

On solving general reverse convex programming problems by a sequence of linear programs and.The book evolved from the earlier book of the author [BNO03] on the subject (coauthored with A.

Nedi´c and A. Ozdaglar), but has diﬀerent character and objectives. The book was quite extensive,wasstruc-tured (at least in part) as a research monograph, and aimed to bridge the gap between convex and nonconvex optimization using concepts.() A finite cutting plane method for solving linear programs with an additional reverse convex constraint.

European Journal of Operational Research() On the global minimization of a convex function under general nonconvex constraints.