Search

Recently classes of conic and discrete conic functions were introduced. In this paper we use the term convic instead conic. The class of convic functions properly includes the classes of convex functions, strictly quasiconvex functions and the class of quasiconvex polynomials. On the other hand, the class of convic functions is properly included in the class of quasiconvex functions. The discrete convic function is a discrete analogue of the convic function. Recently the lower bound $3^{n-1}\log (2 \rho-1)$ for the number of calls to the comparison oracle needed to find the minimum of the discrete convic function defined on integer points of some $n$-dimensional ball with radius $\rho$ was obtained. But the problem of the existence of a polynomial (in $\log\rho$ for fixed $n$) algorithm for minimizing such functions has remained open. In this paper, we answer positively the question of the existence of such an algorithm. Namely, we propose an algorithm for minimizing discrete convic functions that uses $2^{O(n^2 \log n)} \log \rho$ calls to the comparison oracle and has $2^{O(n^2 \log n)} \mbox{poly }(\log \rho)$ bit complexity.

We study the proximity of the optimal value of the m-dimensional knapsack problem to the optimal value of that problem with the additional restriction that only one type of items is allowed to include in the solution. We derive exact and asymptotic formulas for the precision of such approximation, i.e. for the infinum of the ratio of the optimal value for the objective functions of the problem with the cardinality constraint and without it. In particular, we prove that the precision tends to 0.59136.../m if n tends to infinity and m is fixed. Also, we give the class of the worst multi-dimensional knapsack problems for which the bound is attained. Previously, similar results were known only for the case m=1., Comment: 7 pages submitted to MOTOR 2020

In this paper, we consider the class of quasiconvex functions and its proper subclass of conic functions. The integer minimization problem of these functions is considered in the paper, assuming that an optimized function is defined by the comparison oracle. We will show that there is no a polynomial algorithm on $\log R$ to optimize quasiconvex functions in the ball of integer radius $R$ using only the comparison oracle. On the other hand, if an optimized function is conic, then we show that there is a polynomial on $\log R$ algorithm. We also present an exponential on the dimension lower bound for the oracle complexity of the conic function integer optimization problem. Additionally, we give examples of known problems that can be polynomially reduced to the minimization problem of functions in our classes., Comment: Some new proofs have been added. Some fixes are done