<p>Consider the problem of minimizing a lower semicontinuous, semialgebraic function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f :\mathbb {R}^n \rightarrow \mathbb {R} \cup \{\infty \}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> <mo>∪</mo> <mrow> <mo stretchy="false">{</mo> <mi>∞</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {R}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. In this paper, we first show that the set of tangency values (at infinity) of <i>f</i> is a finite set. Then we present necessary and sufficient conditions for the existence of optimal solutions of the problem as well as the boundedness from below and coercivity of the objective function. Finally, we obtain a full characterization of the infimum value at infinity of <i>f</i> and show some relationships between Łojasiewicz exponents at infinity of <i>f</i> for this value.</p>