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In the present paper, we estimate the covering number for some reproducing kernel Hilbert spaces on the unit sphere. Both the upper bounds and the lower bounds are provided.

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<p class="p0">We&nbsp;study&nbsp;the&nbsp;least&nbsp;square&nbsp;regression&nbsp;with&nbsp;data&nbsp;dependent&nbsp;hypothesis&nbsp;and&nbsp;coefficient&nbsp;regularization&nbsp;algorithms&nbsp;based&nbsp;on&nbsp;general&nbsp;kernel.&nbsp;An&nbsp;explicit&nbsp;expression&nbsp;of&nbsp;the&nbsp;solution&nbsp;of&nbsp;this&nbsp;kernel&nbsp;scheme&nbsp;is&nbsp;derived.&nbsp;Then&nbsp;we&nbsp;provide&nbsp;a&nbsp;sample&nbsp;error&nbsp;with&nbsp;a&nbsp;decay&nbsp;of &lt;o(1/sqr(m))&nbsp;and&nbsp;estimate&nbsp;the&nbsp;approxi&nbsp;-mation&nbsp;error&nbsp;in&nbsp;terms&nbsp;of&nbsp;some&nbsp;kind&nbsp;of&nbsp;K-functional.</p>

A kind of tangent derivative and the concepts of strong and weak * pseudoconvexity for a set-valued map are introduced. By the standard separation theoren~ of the convex sets and cones the optimality Fritz John condition of set-valued optimization under Benson proper efficiency is established, its sufficience is discussed. The form of the optimality conditions obtained here completely tally with the classical results when the setvalued map is specialized to be a single-valued map.